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MODERN PIANO TUNING

AND

ALLIED ARTS

INCLUDING

Principles and Practice of Piano Tuning, Regulation of Piano Action, Repair of the Piano, Elementary Princi- ples of Player-Piano Pneumatics, General Construction of Player Mechanism, and Repair of Player Mechanism

BY

WILLIAM BRAID WHITE

Technical Editor of the Music Trade Review. New York. Author of "Theory and Practice of Pianoforte Build- ing." "The Player-Piano Up-to-date," and other works

WITH DRAWINGS. DIAGRAMS. TA^ES. NOTES AND AN INDEX

NEW YORK EDWARD LYMAN BILL, Incorporated

1917

Copyright, 1917, by EDWARD LYMAN BILL, Incorporated

Entered at Stationers' Hall

Murta

LfbrBxy

TO THE CONFERENCE OF

AMERICAN PIANO TECHNICIANS

MEETING IN CHICAGO, U. S. A.

Whose valuable and exhaustive discussions

mark an epoch in the development of

American musical technology.

This Book is, by one who has the honor of membership in that Conference,

RESPECTFULLY AND AFFECTIONATELY DEDICATED

'f^.

PREFATORY NOTE

In writing this book, I have tried to do two things which are always thought to be admirable but seldom thought to be conjunctible. I have tried to set forth the theory of Equal Tempera- ment in a manner at once correct and simple. Simultaneously I have tried to construct and ex- pound a method for the practical application of that theory in practical tuning, equally correct, equally simple and yet thoroughly practical.

The construction of the piano has not in this volume been treated with minuteness of detail, for this task I have already been able to perform in a former treatise ; but in respect of the sound- board, the strings, the hammers and the action, the subject-matter has been set forth quite elaborately, and some novel hypotheses have been advanced, based on mature study, research and experience. Here also, however, the theoretical has been justified by the practical, and in no sense have I yielded to the temptation to square facts to theories,

111

iv Prefatory Note

In the practical matters of piano and player repairing, I have presented in these pages the results of nineteen years' practical and theoreti- cal work, undertaken under a variety of condi- tions and circumstances. In writing this part of the volume I have had the inestimable advan- tage of the suggestions and experiences of many of the best American tuners, as these have been gathered from past numbers of the Music Trade Review, the Technical Department of which pa- per I have had the honor to edit and conduct, without intermission, for fourteen years.

The preliminary treatment of the Acoustical basis of piano tuning may seem elaborate; but I have tried to handle the subject-matter not only accurately but also simply; and as briefly as its nature permits. The need for really accurate in- formation here justifies whatever elaboration of treatment has been given.

I desire here to express my thanks to Mr. J. C. Miller for permission to utilize some of his valuable calculations, to Mr. Arthur Lund, E. E., for drawings of acoustical curves, and to my brother, Mr. H. Sidney White, M. E., for dia- grams of mechanical details.

Most books intended for the instruction and

Prefatory Note v

guidance of piano tuners have been either so theoretical that their interest is academic purely; or so superficial that accuracy in them is through- out sacrificed. I have tried to avoid both er- rors, and to provide both a scientifically correct text-book for teaching and a pocket guide for the daily study and use of the working tuner. The program has been ambitious ; and I am con- scious, now that the task is finished, how far short of perfection it falls. But I think it fills a want ; and I ask of all practitioners and students of the noble art of tuning their indulgence to- wards its faults and their approval of any vir- tues it may appear to them to possess.

The writing of the volume began in the winter of 1914 and was completed during the spring of 1915. Various causes have operated, how- ever, to retard its publication; notably the sud- den passing of the honored man whose en- couragement and kindness made possible the publication of the other books which have ap- peared over my name. It is however fortunate that the successor of Colonel Bill, the corporation which now bears his name and is carrying on so successfully his fine work, has been equally de- sirous with me, of pushing the book to publica-

vi Prefatory 'Note

tion. A thorough rereading of the manuscript, however, during the interim, has suggested many slight changes and a number of explanatory notes, which have been incorporated with, or appended to, the text.

A new, and I hope valuable, feature is the In- dex, which I have tried to make copious and use- ful, to the student and to the tuner alike.

William Braid White. Chicago, 1917.

ERRATUM

Pade 300. For ^Sectional View of Doutle-valve Action ' read ^Sectional View oi Smgle-valve

Action.

OMIT tke following words:

5a Secondary Poucli. r\ V

7a Secondary Reduced Pressure Chamber.

8a Secondary Valve.

11 Primary-Secondary Channel.

Contents

PAGE

Prefatory Note iii

Chapter I. Mechanics of the Musical

Scale ....'.. 1

On the Vibration of a Piano

String 36

Temperament .... 72

Chapter II.

Chapter Chapter

III. IV.

Chapter V.

Practical Tuning in Equal Temperament .... 95

Mechanical Technique of Tuning 114

Chapter	VI.	The Modern Piano .	130
Chapter	VII.	Sound-Board and Strings	152
Chapter VIII.	The Action and Its Regula- tion	184
Chapter	IX.	The Hammer and Its Rela- tion to Tone ....	223
Chapter	X.	Repair of the Piano .	244
Chapter	XI.	Elementary Pneumatics	261

Contents

PAGE

Chapter XII. General Construction of

Player Mechanisms . . 284

Chapter XIII. Eepair of Player Mech- anism 310

Bibliographical Note 329

Index 331

Chapter I.

MECHANICS OF THE MUSICAL. SCALE.

He who undertakes to master the art of piano tuning must have some acquaintance, exact rather than comprehensive, with that general body of knowledge known as Acoustics. This term is used to designate the Science of the phenomena known as Sound. In other words, by the term Acoustics we mean the body of facts, laws and rules which has been brought together by those who have sys- tematically observed Sound and have collected their observations in some intelligible form. Piano Tuning itself, as an Art, is merely one of the branches of Practical Acoustics ; and in order that the Branch should be understood it is neces- sary to understand also the Trunk, and even the Root.

But I might as well begin by saying that no- body need be frightened by the above paragraph. I am not proposing to make any excursions into realms of thought too rarefied for the capacity of

the man who is likely to read this book. I sim-

1

2 Modern Piano Tuning.

ply ask that man to take my word for it that I am going to be perfectly practical and intelligible, and in fact shall probably make him conclude that he has all along been a theorist without knowing it; just as Moliere's M. Jourdain discovered that he had been speaking prose all his life without knowing it. The only difference has been that my reader has not called it "theory." He has called it ''knowing the business."

Anyhow, we are going to begin by discovering something about Sound. We are in fact to make a little excursion into the delectable kingdom of Acoustics.

What is Sound? When a street-car runs over a crossing where another line intersects, we are conscious of a series of grinding crashes exceed- ingly unpleasant to hear, which we attribute per- haps to flat tires on the wheels or to uneven lay- ing of the intersecting trackage. The most prominent feature of such a series of noises is their peculiarly grating and peculiarly spasmodic character. They are on the one hand discontinu- ous, choppy and fragmentary, and on the other hand, grating, unpleasant to the hearing, and to- tally lacking in any but an irritant effect. These are the sort of sounds we speak of as ''noise."

Mechanics of the Musical Scale. 3

In fact, lack of continuity, grating effect and gen- eral fragmentariness are the distinguishing fea- tures of noises, as distinguished from other sounds.

If now we listen to a orchestra tuning up roughly off-stage, the extraordinary medley of sounds which results, may — and frequently does — have the effect of one great noise; although we know that each of the single sounds in the up- roar is, by itself, musical. So it appears that noises may be the result of the chance mixture of many sounds not in themselves noises, but which may happen to be thrown together without system or order. Lack of order, in fact, marks the first great distinction between noises and other sounds.

If now we listen to the deep tone of a steamer's siren, or of a locomotive whistle, we are conscious of a different kind of sound. Here is the im- mediate impression of something definite and con- tinuous, something that has a form and shape of its own, as it were, and that holds the same form so long as its manifestation persists. If, in fact, we continue to seek such sounds, we shall find that what are called Musical Sounds are simply more perfect examples of the continuity, the order and

4 Modern Piano Tuning.

the definite character which we noticed in the lo- comotive whistle's sound. The more highly per- fected the musical instrument, the more perfectly will the sounds evoked by it possess the qualities of continuity, order and definite form.

Continuity, persistence and definiteness, then, are the features which distinguish Musical Sounds^ from Noises. And there are therefore only two kinds of sounds: musical sounds and noises.

Now, what is Sound? The one way in which we can know it, plainly, is by becoming conscious of what we call the Sensation of Sound; that is, by hearing it. If one considers the matter it be- comes plain that without the ability to hear there would be no Sound in the world. Sound cannot exist except in so far as there previously exist capacities for hearing it. The conditions that produce Sound are obviously possible, as we shall soon see, to an interminable extent in all direc- tions ; yet what we may call the range of audible Sound is very small indeed. We can hear so very little of the conceivably bearable material; if I may use so rough an expression.

So it becomes quite plain that Sound cannot be considered as something in itself, existing in the sounding body apart from us, but must rather be

Mechanics of the Musical Scale. 5

thought of as the form in which we perceive some- thing ; the form, in fact, in which we perceive the behavior of certain bodies, which behavior could not be perceived in any other way. Soun(i then can be considered only from the view-point pf the physical laws which govern the behavior of the bodies in question. The laws which govern that sort of behavior which we perceive as Sound, alone form the subject of Acoustics. Why we should experience these perceptio||^ as Sound rather than as Light or Heat is m>t a, question to be decided by/ Acoustics ; is nof a j^oblem of the natural sciences, but of Metaphysics.

Limited th^efore to a strictly mechanical in- vestigation, let us consider the production of Sound from this view-gpf6int. Suppose that I strike a tuning-fork aga*nst the knee and hold it to the ear. I am conscious of a sound only mod- erate in intensity but of persistent and quite defi- nite character, agreeable, and what we call ** musi- cal." No one has any hesitation in calling this a "musical sound." But what produces it, physi- cally speaking? We can discover this for our- selves by making a simple experiment.

By lightly touching the prongs of the fork while it is sounding I discover them to be in a state of

6 Modern Piano Tuning.

vibration. If I examine them under a micro- scope I shall perhaps be able to detect an exceed- ingly rapid vibratory motion. In order however to make sure of the existence of these unseen vi- brations, it is only necessary to obtain a sheet of glass and smoke one surface of it by passing it over the flame of a candle. Then let a tuning fork be fitted with a very light needle point stuck on the end of one prong with a bit of wax, in such

Figure 1.

a position that if the sheet of glass be placed parallel with the length of the fork, the needle point will be at right angles to both.

Now set the fork to sounding, and hold it so that the needle point lightly touches the smoked surface. Have a second person then move the sheet of glass lengthwise while the fork is held still. At once the needle-point will trace out a continuous wavy line, each wave being of that pe- culiar symmetrical form known technically as a

Mechanics of the Musical Scale. 7

curve of sines or sinusoidal curve. By adjusting the experimental apparatus with sufficient exact- ness it would be possible to find out how many of these little waves are being traced out in any given time. Each of these waves corresponds to one vibration or pendulum-like back and forth mo- tion of the fork. By examining the wavy line with close attention, we shall see that if the motion of the glass sheet has been uniform, each sinusoid is identical in size with all the others, which in- dicates that the vibrations are periodic, that is to say, recur at regular intervals and are of similar width or amplitude.

We may therefore conclude from this one ex- periment that the physical producer of musical sound is the excitation of the sounding body into periodic vibrations.

Listen to the noise of the macliinery in a saw mill. When the circular saw starts to bite at a piece of wood you hear a series of grating cracks, which almost instantly assume the character of a complete definite musical sound, though rough in character. As the saw bites deeper into the wood the sound becomes first lower, then higher, until it mounts into a regular song. As the saw comes out through the wood the sounds mount quite high

8 Modern Piano Tuning.

and then instantly die away. What is the cause of this phenomenon?

The circular saw is a steel wheel with a large number of teeth cut in its circumference. Sup- pose there are fifty such teeth. At each revolu- tion of the wheel, then, each tooth will bite the wood once. If the wheel revolves at the rate of say four revolutions per second, it follows that there will be four times fifty or two hundred bites at the wood in this time. That means that the wood will receive two hundred separate scrapes per second. Hence, the rotation of the wheel will be slightly interrupted that number of times in one second. Hence, again, the surface of the air around the wheel will be vibrated back and forth just as many times, because the entry and emer- gence of each tooth will cause an alternate com- pression and suction on the air around it. Try another experiment. Stand five boys up in a row one behind the other, so that each boy has his out- stretched hands upon the shoulders of the boy in front of him. Push the last boy. He falls for- ward, pushes the next and regains his position. Next falls forward, pushes Third and regains his position. Third falls forward, pushes Fourth and regains his position. Fourth falls forward,

Mechanics of the Musical Scale. 9

pushes Fifth and regains his position. Fifth has no one in front of him and so falls forward with- out being able to regain his position. In this way we illustrate the compression and rarefaction of the air by the alternate fallings forward and re- gainings of position undertaken by the boys. The air is even more elastic than the boys and so forms these waves of motion which we saw traced out by the stylus on the tuning fork.

Now, it is plain that as the rotation of the cir- cular saw increases in speed the pulses become suf- ficiently rapid to fuse into one continuous musical sound. If the saw were rotated at irregular, con- stantly shifting speed, the separate shocks would not coalesce and we should have merely the sen- sation of a discontinuous, fragmentary, grating series of shocks which we should call a noise. Thus again we see that regularly recurring mo- tions of the sounding body are requisite to pro- duce musical sounds.

Transmission of Sound. But the illustration of the five boys (which is due to the late Professor Tyndall, by the way) shows something further. It shows first how the excitation of a body into vibration at regular intervals produces an effect upon the immediately surrounding air, causing it

10

Modern Piano Timing.

in turn to oscillate back and forth in pulses of alternate compression and rarefaction. But it shows more. It shows that the sound-motion, as we may call it, is transmitted any distance through the air just as the shock started at one end of the row of boys is felt at the other end, although each boy moves only a little and at once recovers

Figure 2.

his position. So also each particle of air merely receives its little push or compres- sion from the one motion of the tuning-fork or string, and transmits this to the next one. At the backward swing of the tuning-fork or string the air particle drops back to fill up the partial vacuum it left in its forward motion, whilst the motion transmitted to the second particle goes on to the third and to the fourth and so on to the ear of the hearer. Yet each particle has merely os- cillated slightly back and forth. Now, this mode of transmission evidently de-

Mechanics of the Musical Scale. 11

pends upon the existence of an atmosphere. In fact, we can soon show that, apart from all ques- tion of ears, Sound could not exist for us, as we are in this state of existence, without an atmos- phere. Let an alarm-clock be set to ringing and then placed under the glass bell of an air-pump. We now begin to displace the air therefrom by working the handle of the pump. As the quantity of air inside the bell thus becomes smaller and smaller, the sound of the alarm-clock's ringing becomes fainter and fainter, until, where the air is at a certain point of rarefaction, it entirely disappears; although the clapper of the alarm will still be seen working. In other words, there must be an atmosphere or other similar medium, like water, for transmission of the sound-motion from the excited body to the ear.

Properties of Musical Sounds. Having arrived at this point, we are now in a position to discuss musical sounds in general and to discover the laws that govern their behavior. The first prin- ciple we shall lay down is that musical Sounds are distinguished from noises by the continuity of their sensation ; or in other words, musical sounds are evoked by periodic vibrations. It is thus pos- sible to measure the frequency of vibration that

12 Modern Piano Tuning.

evokes a sound of some given Leight; in other words to determine its pitch.

It is also possible, as we shall see, to determine a second quality of musical sounds ; namely, their relative loudness or softness, or, as we shall call it, their intensity.

Lastly, we can discover differences in character or quality between musical sounds, and we shall see also that it is possible to measure these dif- ferences accurately.

Loudness. Let us begin with the second qual- ity mentioned; that of loudness or intensity. If a tuning-fork be excited by means of a violin bow and then examined through a microscope while its motion persists, it will be observed that as the sound dies away, the amplitude or width of swing of the prongs is becoming less and less, until the cessation of motion and of the sound occur to- gether. If, whilst the sound is thus dying away, the fork is again bowed, the amplitude of the prong's motion again is seen to increase just as the sound increases. In fact, it has been found by authoritative experiments that not only does the loudness of a sound vary with the amplitude of the vibrations of the sounding body; but ex- actly as the square of the amplitude. For in-

Mechanics of the Musical Scale. 13

stance, if a piano string can be made to vibrate so that the width of swing in its motion is one-fif- tieth of an inch, and if another piano string giv- ing the same pitch can be made to vibrate with an amplitude of one twenty-fifth of an inch, then the second will have an amplitude twice that of the first and its sound will be four times as loud.

However, let it be remarked that the mechanical operations thus described do not necessarily cor- respond with what we actually seem to hear. In other words, the sensation of loudness and the mechanical cause thereof do not always agree, for the reason that we do not hear some musical sounds as well as others. For instance, it is well known that low sounds never seem as loud as high sounds, even though the amplitude of vibration in each case be the same. A low sound always sounds softer than it really should be, to use a rough expression, and a high sound louder than it really should be.

There is only one more important point about sound-intensity, namely, that the loudness of a sound varies inversely as the square of the dis- tance of the sounding body from the hearer. Thus, other things being equal, a sound heard at a distance of fifty feet should be four times as

14 Modern Piano Tuning.

loud as one heard at a distance of twice fifty, or one hundred feet. However, it must also be re- membered that the situation of the sounding body and of the hearer in proximity to other objects, has a modifying effect upon the loudness of sound as perceived. In fact, we shall see that this is only part of the truth expressed in the term "res- onance," about which we shall have something to say later on.

Pitch. Without making any special attempt at producing an ideal definition of "pitch," it will be enough to call it the relative acuteness or gravity of a musical sound. Everybody knows what is meant by saying that a musical sound is liigh or low. The province of Acoustics lies in finding some measuring-rule, some standard, whereby we can measure this lowness or highness of a sound and place it accurately in relation to all others. The whole system of music is built upon simply a measure of pitch, as we shall see.

Now, first of all, let us find out what it is that makes a sound high or low. In other words, what is the mechanical reason for a sound producing a sensation of highness or lowness?

Musical sounds are produced through the pe- riodic continuous vibration of some body. In the

Mechanics of the Musical Scale. 15

experiment of the circular saw, to which I di- rected attention some pages back, it was pointed out that as the speed of the saw increases, so the musical sound produced through its contact with the wood rises in height. This may be veri- fied by any number of experiments that one chooses to make, and the net result is the fact that the pitch of musical sounds depends upon the number of vibrations in a given unit of time per- formed by the sounding body. Let us put it in a formula, thus:

The pitch of a musical sound varies directly as the number of vibrations per unit of time per- formed by the sounding body: the greater the number of vibrations, the higher the pitch.

Unit of Time. It is customary to assign the second as the unit of time in measuring frequency of vibrations, and in future we shall use this al- ways. If, therefore, we speak of a certain pitch as, say, 500, we shall mean 500 vibrations per second.

Double Vibrations. In counting vibrations, we understand that a motion to and fro constitutes one complete vibration. A motion to or fro would be merely a semi-vibration or oscillation. In the United States and England it is customary to im-

16 Modern Piano Tuning.

ply a double vibration (to and fro) when speak- ing of a ''vibration." In France the single or semi-vibration is the unit of measurement, so that the figures of pitch are always just double what they are as reckoned in the English or American style.

Range of Audibility. It is found as the result of experiment that the human sense of hearing is distinctly limited. The lowest tone that can be distinctly heard as a musical sound is probably the lowest A (A-i) of the piano which, at the stand- ard international pitch, has a frequency of 27.1875 vibrations per second. Sounds of still lower fre- quency may perhaps be audible, but this is doubt- ful, except in the cases of persons specially trained and with special facilities. In fact, any spe- cific musical sounds lower than this probably do not exist for human beings, and when supposed to be heard, are in reality not such sounds at all, but upper partials thereof.^ The 64-foot organ pipe, which has occasionally been used, nominally real- izes tones lower than 27 vibrations per second, but these are certainly not audible as specific separate sounds. They can and do serve perhaps as a bass to reinforce the upper partials of the pipe or the

1 See Chapter II,

Mechanics of the Musical Scale. 17

upper tones of a chord; but they do not appear as separate sounds, simply because the ear does not realize their pulses as a continuous sensation, but separates them. In fact, we may feel safe in concluding that the lowest A of the piano is the lowest of musical sounds generally audible. This statement is made in face of the fact that the sound evoked by the piano string of this note is usually powerful and full. This only means, how- ever, that the sound we hear on the piano is not the pure fundamental vibration of 27.1875 vibra- tions per second, but a mixture of upper partials re-inforced by the fundamental. Of these par- tials we shall have to speak later, for they are of vital importance to the due consideration of our subject-matter.^

A similar limitation confronts us when we come to the highest tones audible by the human ear. Plere again there is considerable diversity of opinion as well as of experience. The highest note of the piano, C7, has a frequency of 4,138.44 vibrations per second at the international pitch. -.J^ However, there is no special difficulty in hear-

1 For a very interesting discussion of the whole question of deepest tones, I refer the reader to Helmholtz, "Sensations of Tone," third English edition. Chapter IX.

18 Modern Piano Tuning.

ing sounds as much as two octaves higher, or up to 16,554 vibrations per second. Above this limit, comparatively few people can hear anything, al- though musicians and acousticians have been able to go much higher.^

The Musical Range. The limits of audibiUty therefore embrace eleven octaves of sounds, but the musical range is considerably smaller. The modern piano embraces virtually the complete compass of sounds used in music, and, as we all know, that range is seven octaves and a minor third, from A-i to C7.

Let it be noted that if the range of bearable sounds lies between, say 27 and 32,000 vibrations per second, the number of possible distinct musi- cal sounds is enormous. We know that it is quite possible for the trained ear to discriminate be- tween sounds which, at the lower end of the gamut anyhow, are no more than 4 vibrations per second apart. For many years the late Dr. Rudolph Koenig of Paris, one of the most gifted acousti-

1 Many years ago, before I had become practically interested in Acoustics, and when my ear therefore was in every sense untrained, I was tested by the Galton whistle up to 24,000 vibrations per second, which is near G^, two and one-half octaves above the piano's hif^hest note. This is well up to the higher limit of most trained ears, although some acousticians have tuned forks run- ning up to Cjo, with 33,108 vibrations per second.

Mechanics of the Musical Scale. 19

cians the world has ever known, was engaged in the construction of a so-called Universal Tono- meter, consisting of a superb set of one hundred and fifty tuning forks, ranging in frequency from 16 to 21,845.3 vibrations per second. In this re- markable instrument of precision, the lowest sounds differ from each other by one-half a vi- bration per second, while within the musical com- pass the difference never exceeds four vibrations. It can readily be seen therefore that the number of possible musical sounds is very much greater than the eighty-eight which comprise the musical gamut of the piano.

Just how the musical scale, as we know it, came to be what it is, I cannot discuss here; for the simple reason that the whole question is really to one side of our purpose.^ Whatever may be the origin of musical scales, however, we know that the diatonic scale has existed since the twelfth cen- tury, although the foundation of what we call mod- ern music, employing the chromatic tempered

1 The claims made for the eleventh century monk, Guide d'Arezzo, have been disputed, and the reader who is interested in the historical aspect of the subject is referred to Grove's "Dic- tionary of Music and Musicians," to Helraholtz' "Sensations of Tone," and to A. J. Ellis' "History of Musical Pitch," quoted in Appendix 20 of his translation of Helmholtz (3rd edition).

20 Modern Piano Tuning.

scale, was rightly laid only by Sebastian Bach, who died 1750. Music is a young, an infantile, art, as time goes.

The Diatonic Scale. We have already seen that the musical tone is a fixed quantity, as it were, being the sensation that is produced or evoked by a definite number of vibrations in a given time. This being the case, it becomes evi- dent that all possible tones must bear mathemati- cal relations to each other. As long ago as the sixth century b. c. the Greek philosopher and scientific investigator Pythagoras propounded the notion that the agreeableness of tones when used with each other is in proportion to the simplicity of their mathematical relations. Now, if we look at the scale we use to-day we find that although the relations of the successive members of it to each other appear to be complex, yet in fact these are really most simple. Let us see how this is :

Unison. We all know that we can recognize one single tone and remember it when we hear it a second time. If now we draw the same tone from two sources and sound the two tones together, we find that they blend perfectly and that we have what we call a Unison. If we were to designate

Mechanics of the Musical Scale. 21

the first tone by the mathematical symbol 1, we should say that the Unison is equivalent to the proportion 1 : 1. This is the simplest of rela- tions ; but it is so because it is a relation between two of the same tone, not between two different tones.

Octave. We all recognize also the interval which we call the octave and we know that in reality two sounds an octave apart are identical, except that they exist on different planes or levels. So, if we play the sound C and then evoke the C which lies an octave above, we find that we have two sounds that actually blend into one and are virtually one. When we come to discover the re- lations between two sounds at the octave inter- val, we find that the higher sound is produced by just twice as many vibrations in a given time as suffice to produce the lower sound, and so we can express this octave relation mathematically by the symbol 1 : 2. This is a relation really as simple as that of the Unison, for in reality the Octave to a given tone is simply a Unison with one member thereof on a higher or lower plane.

Perfect Fifth. The relation next in simplicity should naturally give us the next closest tone-re- lation. And we find that the ear at once accepts

22 Modern Piano Tuning.

as the next closest relation what is called the Fifth. If one strikes simultaneously the keys C — G upwards on the piano one observes that they blend together almost as perfectly as the tones C — C or G — G, or any other octave or unison. The Interval or relation thus sounded is called a Perfect Fifth. When we come to trace up its acoustical relations we find that a tone a Fifth above any other tone is produced by just one and a half times as many vibrations in a given time as suffice to produce the lower tone. Thus we can place the mathematical relation of the interval of a Perfect Fifth as 1 : IH, or better still, for the sake of simplicity, as 2 : 3, which is the same thing. So we now have the simplest relation that can ex- ist between different tones; the relation of the Per- fect Fifth or 2:3. This important fact will lead to essential results, as we shall see.

The Natural Scale. This interval, the Fifth, will be found competent to furnish us with the en- tire scale which the musical feeling and intuition of men have caused them, throughout the entire Western World at least, to accept as the basis of music and of musical instruments ; that is to say, the diatonic scale. If we begin with the tone C at

Mechanics of the Musical Scale.

23

any part of the compass and take a series of Fifths upwards we shall arrive at the following scale :

C G D

E B F sharp.

JSl

m

ot

"ST

D A

Figure 3,

B

F SHARP

These tones of course are spread over a compass of five octaves, but if they are drawn together into the compass of one octave, as they may rightly be drawn (see supra ''The Octave") then we shall have a scale like this :

C D E F sharp GAB

Now the F sharp in the present case is not ac- tually used, but instead we have F natural, which in fact is drawn from the interval of a Perfect Fifth below the key-tone C. The reason for this preference of F natural over F sharp lies in the

24 Modern Piano Tuning.

fact that the diatonic scale is thereby given a cer- tain symmetry of sound which otherwise it would lack and because the work of practical musical composition is advantaged by the substitution/

The Diatonic Scale. We have arrived now at the Diatonic Major Scale and although we need not here be concerned with the origin thereof, we may be satisfied to know that it appears to sat- isfy the musical needs of civilized mankind. Let us again examine the series of tones, this time including the octave to C, whereby we in reality complete the circle of Fifths, as it may be called, and return to the key-tone, for the octave is the same for musical purposes as the Unison. We have then, counting upwards,

CDEFGABC

which we can readily identify as the series of seven white keys on the piano ; with the eighth fol- lowing and beginning a new series or scale. The complete diatonic scale, when founded on the tone C, may thus be seen, merely by looking at the piano, to consist of a series of such scales, seven

1 For a general discussion of these reasons consult Goetschius' "Theory and Practice of Tone-Relations."

Mechanics of the Musical Scale. 25

in all, following one another from one end of the piano to the other.^

Relations. Now, if we go a step further and discover the relations which these tones hold to each other mathematically, when brought together into one octave, we find them to be as follows, ex- pressing the lower C as 1 and the upper C as 2, and counting upwards always:

CDEFGABC

1 9/8 5/4 4/3 3/2 5/3 ,15/8 2

Or in other words, the relation C to D is the same as the ratio 8 to 9. The relation C to E is like- wise 4 to 5. The relation C to F is 3 to 4, C to G is 2 to 3, C to A is 3 to 5, C to B is 8 to 15 and C to its octave is 1 to 2.

Tones and Semitones. Now if we glance at the C scale as shown on the white keys of the piano we shall see that it exhibits some interesting pe- culiarities. Between each pair of white keys, such as C — D or D — E, is a black key, which most people know is called a sharp or a flat. But between E — F and B — C is no space whatever, these pairs of white keys being immediately adjacent to each

1 Note, however, that the modern piano contains three tones lower than tlie lowest C, making a minor third more of compass.

26 Modern Piano Tuning.

other. If we run over the keys to sound them we shall find that the sound-interval between E — F or B — C can at once be heard as being closer or narrower, as it were, than the sound-interval be- tween A— B or C— D or D— E, or F— G or G— A or A — B. The longer intervals, between which we find the black keys, are called Diatonic Whole Tones, and the shorter intervals E — F and B — C are called Diatonic Semitones.

Diatonic Relationships. The exact relations subsisting between the steps or degrees of the Diatonic Scale can be ascertained by dividing the ratios previously had, by each other, pair to pair. Consulting the table previously given {page 25) showing the relations of the steps to their key- tone, we find that when the ratios are divided pair by pair we get the following relations between each pair of notes :

C D....E..F G....A B..C,

8:9 9:10 15:16 8:9 9:10 8:9 15:16

Now the first thing that will be observed is that there are three intei'vals here, not two. There are in fact, evidently two kinds of whole-step or whole-tone. For it is evident that the sound-dis- tance between C and D is more than the sound-

■i.4 5

Mechanics of the Musical Scale. 27

distance between D and E. In actual fact, these two whole-steps must be recognized as distinct. This, however, brings about an entirely new con- dition and one quite unsuspected. For inasmuch as the Diatonic Scale must of course always re- tain the same relationships among its successive steps, it is evident that this idea of two ditferent kinds of whole-step must land us in difficulties.

The trouble is that we cannot always play in the key of C, by which I mean that sometimes, in fact very often, we desire to build our music upon Dia- tonic Scales which are founded upon other tones than C. From the point of view towards which I am leading — namely, that of tuning — we see here a serious difficulty, for it is at once evident that if we undertake to tune a Diatonic Scale, as sug- gested some time back, by considering it as a series of Perfect Fifths, we shall find ourselves in deep water as soon as we quit the key of C. Let me make this plainer.

Understand first of all that we have as yet talked only of a scale founded on C and therefore including what are known simply as the white keys or natural notes. Suppose we begin by tuning a series of fifths quite perfect from some given C, say for the sake of convenience a C of which the

28

Modern Piano Tuning.

pitcli is 64 vibrations per second. This is a little less than the pitch of C would be at the Interna- tional standard but is more convenient for pur- poses of calculation.

Then we should get a result like this:

m

-o-

F42.66

C64-

G96 DI44 AEI6

Figure 4.

E324 B486

Now, let us reduce this down to one octave, by transferring the higher tones down, through the simple process of dividing by 2 for each octave of transference down and multiplying by 2 for each octave of transference up. This will give us the folloAving result:

F42.66 C 64.

G96 DI44 A 216

Figure 5.

E324

^^
^ n O
< c	9- 64	-2> ^^. -^ .-'"''-''' n73^ ro,-'^' ''85.31'- ^.G96 AI08 / BI2I.5
w\*	.- ^ ^- '-' ~	y
/•		■jf /
/		(m ^
	-•	r,^	M-' iT)
.	jr-	-		tf ^^^

B486

Mechanics of the Musical Scale. 29

Gathering this together, we have the following scale founded on C = 128, or in acoustical nota- tion Co = 128.1

Co	D,	Eo	F,	G,	A,	B,	Ca
128	144	162	170.66	192	216	243	256

Now, suppose we want to play a tune based on another key-tone than C. Suppose, for instance, that we want to use D 144 as the basis of a scale ; that is to say, we want to play in the key of D, as we say. The first thing to do is to find out whether we have notes tuned already which will give us such a scale. Going on the same plan as the pattern Diatonic Scale of C, and applying it to D, we find that we need the following notes :

D E F# G A B C# D.

All of these we already have except F# and C#. We can get F# by tuning a perfect Fifth above B 243, which will give us FjJ 364.5. Dropping this an

1 Acoustical notation is as follows: Lowest C on the piano is ca:lled C. The second C is C„ the third is Cj, middle C is C3 and so on up to the highest note on the piano, which is C,. The notes between the various C'a are called by the number of the C below. Thus, all notes in the middle C octave, between C, and C^ are called D3, E3, F3, etc., up to C^, when they begin again D^, E4, etc., up to C5. This is the modern notation and I shall use it exclusively.

30 Modern Piano Tuning.

octave we have F# 182.25. C# is a Perfect Fifth above F# and so will be 546.75, or, dropping an octave, 273.375. Now, we can construct a scale of D as follows, beginning with the D 144 that we al- ready have, using all the other notes already pro- vided and the two new ones besides. That gives ust

D,	Eo	F#2	G,	A,	B,	C#3	D3
144	162	182.25	192	216	243	273.375	288

If you will look at it closely you will see that there must be something wrong. The distance be- tween F# and G seems small, and so does the dis- tance between C# and D. To test the thing, let us now construct a diatonic scale on the ratios we know to be correct ^ and see what results we get. It works out as follows :

D2 E^ F#2 G2 A, B, C#3 I>3

144 162 180 192 216 240 270 288

Ratios

8:9 9:10 15:16 8:9 9:10 8:9 15:16

Now, just for purposes of comparison, let us put these two scales together, one below the other. They look like this :

1 See pa{2^e 25 et seq.

Mechanics of the Musical Scale. 31

SCALE MADE UP FROM C SCALE AND PEBFECT FIFTHS TUNED THERE- FROM

D	E	F#	G	A	B	c#	D
144	162	182.25	192	216	243	273.375	288

SCALE MADE UP FROM KNOWN DIATONIC RATIOS

144 162 180 192 216 2JfO 210 288

At once it can be seen that the F# and the C# which we manufactured by the perfectly legitimate method of tuning perfect Fifths from the nearest tone available in the scale of C, are both wrong when secured in this way. Also, it can be seen that the B which belongs to the scale of C will not do for the scale of D. Not only is this so, but if the experiment is made with other key-tones, it will be found that they all, except the scale of G, differ somewhere and to a greater or less extent from the scale of C, even with reference to the notes which they have in common with C

True Intonation. It is evident, therefore, that no method of building up diatonic scales by tuning pure intervals, will do for us if we are going to use the same keys and the same strings for all the scales we need. It is evident, in fact, that if we tune perfect Fifths or any other intervals from C or any other key- tone and expect thereby to gain a scale that will be suitably in tune for all

32 Modern Piano Tuning.

keys in which we may want to play, we shall be disappointed. Not only is this so, but it must be remembered that so far we have not attempted to consider any of the so-called sharps and flats, except in the one case where we found two sharps in the scale of D, properly belonging there. It turns out, however, when we investigate the sub- ject, that the sharp of C, when C is in the scale of C, is quite a different thing, for instance- (as to pitch), from the C# which is the leading tone of the scale of D.

Chromatic Semitone. The chromatic semitone, which found its way into the scale during the formative period of musical art — mainly because it filled a want — is found upon investigation to bear to its natural the ratio ^%5 or 2%4, according as it is a flat or a sharp. In the case we have been considering, then, whilst C# as the leading tone in the scale of D has a pitch, in true intona- tion, of 270, the C# which is the chromatic of C 256 (see previous tables) would have a pitch of 256 X ^%4 or 266.66. Similar differences exist in all cases between chromatic and diatonic semi- tones, thus introducing another element of con- fusion and impossibility into any attempt to tune in true intonation.

Mechanics of the Musical Scale. 33

Derivation of chromatic ratio. Actually the chromatic semitone is the difference between a ^% ratio whole tone or minor tone as it is often called, and a diatonic semitone ; thus ^% -—

The Comma. The difference between the % (major) and the ^% (minor) tones is called a comma = ^%o. This is the smallest musical in- terval and is used of course only in acoustics.

(%-^^% = «yso).^

Musical Instruments Imperfect. The above dis- cussion, then, leads us to the truth that all musical instruments which utilize fixed tones are neces- sarily imperfect. As we know, the piano, the organ and all keyed instruments are constructed on a basis of seven white and five black keys to each octave, or as it is generally said, on a 12-to- the-octave basis (13 including the octave note). If now we are to play, as we of course do play, in all keys on this same key-board, it is evident that we cannot tune pure diatonic scales. The imper- fection here uncovered has, of course, existed ever since fixed-tone musical instruments came into be- ing. The difficulty, which has always been recog-

1 The diatonic minor scale is affected equally by this ar,?iimcnt; but has not been mentioned here for reasons set forth in Chap.

in.

34 Modern Piano Tuning.

nized by instrument builders and musical theorists, can be put succinctly as follows :

The piano and all keyed instruments are imper- fect, in that they must not be tuned perfectly in any one scale if they are to be used in more than that one scale. Hence a system of compromise, of some sort, must be the basis of tuning.

The violins and violin family, the slide trombone and the human voice can of course sound in pure intonation, because the performer can change the tuning from instant to instant by moving his finger on the string, modifying the length of the tube or contracting the vocal chords. When they are played, however, together with keyed instru- ments, the tuning of these true intonation instru- ments is of course modified (though unconsci- ously), to fit the situation.

All tuning imperfect. All tuning, therefore, is necessarily imperfect, and is based upon a system called ' ' Temperament. ' ' This system is described and explained completely in the third and fourth chapters of this book.

Temperament. I have taken the reader through a somewhat lengthy explanation of the necessity for Temperament on the notion that thereby he will be able to understand for himself, from the

Mechanics of the Musical Scale. 35

beginning, the necessity for doing things that otherwise would seem illogical and inconsistent. The peculiar kind of tuning that the piano tuner must do would seem in the highest degree absurd if the student did not understand the reasons for doing what he is taught to do. Seeing also that this correct knowledge is seldom given by those who teach the practical side of the art, I thought it better to go into some detail. In any case, it is well to realize that no man can possibly be a really artistic piano tuner unless he does know all that is contained in this chapter and all that is contained in the next three. It is worth while therefore to be patient and follow through to the end the course of the argument set forth here.^

1 A complete discussion of the problem of True or Just Intona- tion is to be found in the classic work of Helmholtz, to which the reader is referred. See especially Chapter XVI, Appendices 17 and 18 and the famous Appendix 20, composed by the English translator, A. J. Ellis. My own "Theory and Practice of Piano- forte Building" contains (Chapter VI) a useful discussion of the Musical Scale and Musical Intonation.

Chapter II.

ON THE VIBRATION OF A PIANO STRING.

Of all sounding bodies known to music, tlie musical string is without doubt the most common, the most easily manipulated for musical and me- chanical purposes, and the most efficient. Ac- quaintance with ascertained facts as to the be- havior of musical strings under practical condi- tions is necessary for the complete equipment of the piano tuner; although this acquaintance need not be exhaustive, so long as it be, to its extent, exact. Avoiding mathematical symbols which, requisite as they are to a comprehensive study of Acoustics, may nevertheless be beyond the famil- iarity of most of those who will read this book, I shall here briefly investigate certain properties of musical strings and especially of the piano string. The discussion, I can promise, need seem neither dry nor uninteresting.

The String. To be exact, a string should be defined as a perfectly flexible and perfectly uni-

36

On the Vibration of a Piano String. 37

form filament of solid material stretched between two fixed points. But such a string, it must be ob- served, can exist only as a mathematical abstrac- tion, since neither perfect uniformity nor perfect flexibility can be expected in strings made by human hands. A string of given flexibility be- comes more flexible, as to its whole length, if that length be increased, and, conversely, stiffer as its length is decreased. The property of weight also fluctuates in the same way. Likewise if the force whereby a given string is stretched between two points be measured in a given number of pounds, the effective tension equivalent thereto will of course be decreased if the string be lengthened ; or conversely will be increased if the string be short- ened. A string 12 inches long stretched with a weight of 10 pounds is subjected to a lower tension than is a 6-inch string of similar density and thickness stretched with the same weight. These allowances and corrections, obvious as they are, must constantly be kept in mind if we are to un- derstand the behavior of practical strings, since all the acoustical laws which govern such behavior must be modified in practice according to the facts disclosed above. Simple Vibration. We have learned that mus-

38 Modern Piano Tuning.

ical sounds owe their existence to the fact that some solid body is thrown into a state of periodic vibration. The kind of vibration can best be ex- plained by likening it to the swing to and fro of s a pendulum. A pendulum is fixed at one end and tends naturally to swing back and forth on its pivot. The kind of vibration which the pendulum performs is called simple or pendular vibration. The tuning fork, when set in vibration, is also very much the same thing as a pendulum, since one end of each prong is fixed and the other end can there- fore swing freely. The tuning fork furnishes, when excited, an excellent practical example of simple or pendular vibration of sufficient rapidity to produce musical sound.

Tones of the Piano String. Go to a piano and strike one of the low bass keys in the octave be- tween A-i and A. These very low keys operate on single strings only and hence are excellently adapted for our purpose.^

Strike on the piano the key (say) F. Hold the

1 Incidentally, let me say that the piano is an almost complete ready-made acoustical instrument for the investigation of the phenomena of musical strings, sympathetic resonance, beats and beat-tones, and partials. With a piano at hand the student can dispense with all experimental means except the tuning-fork. I shall suppose that a piano is at hand during the reading of this and other chapters.

On the Vibration of a Piano String. 39

key down, and listen carefully. At first you will hear simply the full sonorous tone F, deep and solemn. But listen closely, repeating the experi- ment till the ear becomes familiar, and you will gradually observe that, mingled with the original sound F, there are a number of other sounds, ap- parently very closely related to the original, color- ing it rather than altering its pitch, but at the same time recognizable as sounds that spring from a different level. By repeating the experiment with various of these low strings (or by going higher and taking care that one string in the two- string unisons is damped off), you will gradually be able to perceive the remarkable fact that every piano string produces a sort of compound tone, consisting primarily of its natural tone or funda- mental, as we may call it, but containing also the octave thereto, the fifth above that and the second octave. It is true that these extra sounds are feeble and can be heard only by means of practice and the exercise of patience; but heard they can be, more and more clearly as one's familiarity with the process grows.

Partial Tones. The truth is that the piano string does not evolve a simple but an exceedingly complex musical tone. Not only the three extra

40 Modern Piano Tuning,

tones of which I spoke before can be proved to exist, but in fact an immense number of other tones, all bearing given harmonic relations to the fundamental, can be shown to be evoked, and by the use of suitable apparatus can be detected and isolated, one by one, through the sense of hear- ing. Special resonators have been made which enable the hearer to detect these partial tones clearly.

Even without such special apparatus, however, we can detect a number of the partial tones if we take advantage of the piano 's property of sympa- thetic resonance; a property imparted by the sound-board.

Sympathetic Resonance. Hold down the middle C key, without striking the string. Then, while holding the key down, strike a powerful blow on the C immediately below. When the sound has swelled up, let go the lower key whilst holding on to the upper or silently pressed key. At once the sound of middle C floats out of the silence, pure and ethereal. What is the cause of this sound? How has the middle C string been excited? The answer is found in the fact that the lower string which was struck, not only produces its funda- mental tone but also evokes its octave above. The

On the Vibration of a Piano String. 41

peculiar sort of vibration of the C2 string which produced this octave is resonated through the sound-board and reproduced on the middle-C string. In the same way, the twelfth (G3) can be brought out, and so can the next octave C4. In fact, with a very good piano and by choosing a low enough sound for the fundamental, even higher partial tones can thus be brought out by sympa- thetic resonance from the original string to the string corresponding with the true pitch of that partial.^

Complex String Vibration. Thus we learn that the piano string vibrates as a complex of vibra- tions, not as one simple form of vibration; for it is evident that if the string evokes, as we know it does, a complex of sounds, these must arise from a complex of vibrations. Let us see how this is :

Turning again to the piano, select a string in such a position that it can be measured accurately as to its speaking length. A grand piano is most convenient for the purpose, and the string may be selected from the overstrung or bass section. Now accurately measure the speaking length of the string between bridges, and mark carefully

1 For a further discussion of sympathetic resonance, see Chap- ter VII.

42 Modern Piano Tuning.

with a piece of chalk on the sound-board the exact middle point as near as you can determine it. Then sound the string and whilst holding down the key, touch the string at the middle point very lightly with a feather. If you perform the operation skilfully enough, you will find that in- stantly the fundamental tone of the string ceases and there floats out the octave above, quite alone and distinct.

Measure now one-third of the length, mark it, and again sound the string. Placing the feather carefully at the exact division point and damping the shorter segment with a finger, the fifth above the original sound is heard.

Automatic string division. What is the mean- ing of all this? Plainly in the first case it meant that the string naturally subdivides itself into two parts of equal length and that the vibration of either half gives the octave above the original. Thus we have two vitally important facts at our disposal, one relating to the form of vibration of the string and the other to the law of string length as proportioned to pitch.

Moreover, in the second case, if we allowed the Vs division of the string to vibrate, we should get from it a sound an octave above the sound of the

On the Vibration of a Piano String. 43

longer or % division. Since we damped the shorter segment, however, we conclude that the fifth above the original sound was produced by a string length % of the original length. If we now continue our experiments we may find that % of the original length produces a major 3rd above the original sound, and that H of the length produces a sound 2 octaves above the original sound. Plainly then, we have two great laws revealed. The first is :

When a string fixed at each end like the piano string, is struck at one end, it vibrates in a com- plex form, most strongly in its full length but also perceptibly in segments of that length such as ^, %, % and H.

The second law is equally important. It may be stated as follows:

Length and Pitch. The pitch of a string — that is to say, the number of vibrations, per unit of time, it can perform, is proportional inversely to its length. Thus, since an octave above a given sound has twice as many vibrations per second as the original sound, it follows that to obtain a sound an octave above a given sound we must have a string one-half as long.

Weight, Thickness and Tension. Similar laws

44 Modern Piano Tuning.

exist with regard to the influence upon string vi- brations of weight, thickness and tension. With- out undertaking to prove these completely, we may state them briefly as follows :

The frequency of a string's vibration is in- versely proportional to the square root of its weight. In other words, if the weight be di- vided by 4 (the square of 2) the frequency will be multiplied by 2. To produce a tone one octave below its original tone, the weight of the string must be increased in the proportion 4:1. To produce a tone one octave above the original tone, the weight of the string must be only Yi its original weight.

The frequency of string vibrations is directly proportional to the square root of their tension. In other words, to get twice as many vibrations, you must multiply the tension by 2 ^ = 4. To get four times as many vibrations you must multiply the tension by 4 ^ = 16. So if a string be stretched with a weight of 10 lbs. and it is desired to make it sound an octave higher, this can be done by mak- ing the stretching weight (4 X 10)= 40 lbs.

The frequency of string vibrations is inversely proportional to the thickness of the string. If a string of a given length and weight produces a

On the Vibration of a Piano String. 45

tone of a given number of vibrations, a string of the same length and twice the thickness will give a tone one octave lower; that is, of half the num- ber of vibrations.

Mechanical Variable Factors. All of these laws, be it remembered, are based on the assump- tion of mathematical strings, in which weight and stiffness remain constant through all changes in length. In the case of the actual piano string, in which the weight and tension do vary with the length, some compensation must be made when calculating. Thus, to illustrate, it is found that whereas the acoustical law for frequency of vi- bration requires a doubling of the string length at each octave downwards or halving at each octave upwards, the practical string, where weight and tension vary with length, requires a proportion of 1 : 1.875 instead of 1 : 2. This difference must be kept in mind.^

Why Strings Subdivide. Before, however, we go on to consider the influence exerted, through the peculiar manner in which strings vibrate, upon the problems of tuning and tone-quality, we must take the trouble to discover why they should, in

1 For discussions of this point, see my "Theory and Practice of Pianoforte Building," Chapter VIII.

46 Modern Piano Tuning.

fact, vibrate in this rather than in some other way. It is easy to talk about string subdivision and partial tones, but there is very little use in mouth- ing words that do not carry with them to our minds real meanings, or in talking about processes which we do not really understand. So, let us take the trouble to discover why a string vibrates as we have shown it to vibrate. Here again, the piano shall be our instrument of investigation.

The Wash-line Experiment. The first thing to realize when we begin to talk about string vibra- tions is that the vibration itself is merely the transmission of a motion from one end of the string to the other. This motion will continue un- til it is transformed into some other sort of energy or else is thrown out of its direction into another direction. Suppose that you take a long cord, like a wash-line. Obtain one as much as twenty feet long. Fasten one end to a post and stretch the cord out until you hold the other end in your hand with the entire length fairly slack. Now try to jerk the cord up and down so that you can get it to vibrate in one long pulse. That pulse will af- fect the entire length of the cord, which you will observe to rise from its plane of rest, belly out in a sort of wave, descend to the point of rest

On the Vibration of a Piano String. 47

again, belly out once more on tlie opposite side and return to the point of rest, making a complete swing to and fro. Compare the illustration fig- ure 6.

When you find that you can do this (practice is needed), try a different experiment. Try to jerk the cord with a sort of short sharp jerk so that, instead of vibrating in its whole length, a sort of

Figure 6.

hump is formed on the cord which travels like a wave through a body of water. This short wave will travel along the whole cord, as you will be able to see by watching it narrowly, until it comes to the end fixed on the post. At once you will see that the wave, instead of disappearing, is reflected back, reversing its direction of travel and also its position, being now on the opposite side of the cord. Thus reflected, the short wave travels back to you.

This is an example of what is called the reflec-

48 Modern Piano Tuning.

Hon of a sound impulse. But it has a very impor- tant bearing on the general problem of piano string vibration, as we shall see.

Suppose that you are able to time your efforts so carefully that you can deliver a series of these short sharp jerks, forming these short waves, at the rate of one per second. If you time your im- pulses carefully, you will find that the second impulse will start away from your hand just as the

Figure 7.

first impulse starts back from the fixed end of the string. The two impulses, traveling in opposite directions and in opposite phases of motion (in positions on the cord opposite to each other), will meet precisely in the center, for neither one can pass the other. At their meeting place, the exact middle of the string, the forces are equal and opposite, so that a node or point of greatly dimin- ished amplitude of motion is formed. The two pulses therefore have no option but to continue vibrating independently, thus dividing the string into two independently vibrating halves, each of

On the Vibration of a Piano String. 49

double the original speed. See the illustration figure 7.

Meanwhile a second impulse from the hand be- gins to travel along the cord and upon its meeting the already segmented halves the result is a fur- ther reflection and subdivision. This again con- tinues still further at the next impulse, so that finally, if the impulses can be kept up long enough, the result will be the division of the cord into four, five, six, and up to perhaps ten of these "ventral segments," separated by nodes.

Harmonic Motion. This being the mode of vi- bration of slow moving cords, we can see how the rapidly moving piano strings are instantly thrown into the state of complex vibration described above ; for we must remember that not only is the vibration very rapid, ranging from 27 to more than 4100 vibrations per second, but also that there is no limit to the possible number of subdivisions. Moreover, the piano string is very stiff and being fixed at both ends and excited by a stiff blow, its motion is not only rapid but powerful, so that the reflections are unusually strong and numerous. Hence the wave of motion of the piano string is remarkably complex.

Besultant Motion. The entire complex motion

50

On the Vibration of a Piano String. 51

of the piano string is of course the result of the operation of many forces, moving from different directions, upon a single resistance; so that the result of the interference of the forces with each other is that their net efficiency works out in some direction which is a resultant of all the directions. Thus, the piano string, if it be examined under mo- tion by any optical method, is seen to vibrate in a wave motion which is the resultant of all the par- tial motions. The general appearance of such a wave is as shown in the illustration, figure 8, which gives (theoretically) the resultant of a wave mo- tion including subdivision into six segments.^ The piano usually has the first eight and often the ninth, in the lower and middle registers, and still higher partials in the high treble, but the latter can hardly be isolated without special apparatus ; and then not easily. Of course, as we shall see later, there are certain causes which affect the form of the wave in the piano string, in practical

1 This illustration is after the original by Prof. A. M. Mayer, of Stevens Institute, one of the most eminent of American acous- ticians. Professor Mayer's drawings of harmonic curves and re- sultants were first published in the Philosophical Magazine for 1875. In order to show clearly the six separate wave motions, their respective amplitudes have been made proportional to wave length. This is of course a scientific fiction, but the eflFect upon the resultant curve is not markedly distorting.

52 Modern Piano Tuning.

conditions, and so modify the series of partials. Fourier's Theorem. One of the greatest of French mathematicians, Fourier, investigating an- other subject altogether, discovered the law of this harmonic motion of a string when he showed that every complex vibratory motion can be re- duced to a series of simple pendular motions, of which the terms are as follows :

1 1/ 1/ 1/ IZ 1/. 1/4 1/ 1/. V-ir, ^i<^- «"^ infinitum. 1, 72, 73, A, /5, /G, /7, /S, /9, /lO,

In other words, the very subdivision of the piano string into segments is here shown mathematically to be the necessary basis of all compound motion in vibrational form. Thus mathematics, from an- other angle, amply confirms the ideas above set forth.i

Partial Tones. The string, then, vibrates in its whole length, its y2, Vs, Vi, M, Yc, Vi, Vs, and smaller segments indefinitely. The whole length vibra- tion produces the fundamental tone of the string. The % gives twice as many vibrations, or the oc- tave above. The Vs gives the twelfth, and the % gives the double octave. Thus, the piano string C 64, when sounded, actually involves not only the fundamental tone but all the following:

ij. B. Fourier, 1768-1830, author of "Analysis of Determinate Equations."

On the Vibration of a Piano String. 53

First 16 partiala of Ci = 64.

Ej 63 Bbj C4 D4 E* Ft»4 G* A* Bb* BK C5

^

?

\>0 y

^

r rf^r f

12 3 4

64 128 192 256

5 6 7 8 9 10 II 12 13 14 15 r6 320 384 448 512 576 640 704 766 832 896 960 (024

Figure 9.

and many more not shown. All these are called partials. The fundamental is the first partial, the octave is the second, and so on. Above the tenth, although the number of possible subdivisions is unlimited, the pitch becomes less and less definitely referable to any specific note of the scale. The first 6 partials, as can be seen, are simply the common chord of C spread out. The 8th, 10th, 12tli and 16th are octaves to the 4th, 5th, 6th and 8th. The 7th is flatter than the diatonic seventh which we use, although the former is the natural tone and the latter quite artificial.^ The other odd-numbered partials are all more or less out of tune with their nominal equivalents, until, at a short distance above the 16th, all pretence of con- sonance except in the 20th, 24th and 32nd, has van- ished.

1 Cf. Chapter 1, "Natural Scale."

54 Modern Piano Tuning.

Influence of Partials. It should be remembered that, although all the odd- and most of the even- numbered partials above the 10th are dissonant and this dissonance progressively increases — if one may use the term — the number of partials that may occur above the 10th in a piano string is quite large. This being the case, it will be under- stood that although these partials are relatively feeble, and their sounds do not affect the general sensation of pitch, they do have another effect; and this is felt in what is called the ** quality" or ''color" of the sound. In fact, as we shall soon see, the harshness or mellowness, thinness or full- ness, of a sound, as evoked from the piano, not to mention the greater characteristic differences which distinguish the tone of one instrument from that of another, are all to be attributed to the manner in which the various partials are mixed with the fundamental.

Series of Partials. But why should there be from one piano string a mixture of partial tones different from that which persists in another? For that matter, since the tendency of other sonor- ous bodies, like pipes for instance, is to divide up naturally into ventral segments, like strings, why should not all tone quality be alike? Obviously

On the Vibration of a Piano String. 55

the difference must arise because one wave form varies from another ; or in other words because one string or pipe or rod produces one specific mix- ture of partials and another a different mixture. Why this should be so in the case of the piano string, which is our present concern, I shall now comprehensively explain, and the following dis- cussion will be of great assistance in promoting an understanding of some most important prob- lems.

Point of Contact. The piano string is excited by a more or less violent blow from a felt-covered hammer. The impulse thus given to the string is relatively powerful, and its effect upon the highly tensioned filament of steel is such as to induce instant reflection of the sound-impulse and sub- division of the string into many ventral segments. But the exact individual segments into which the subdivision takes place are determined by one spe- cial condition ; namely, by the position of the ham- mer's point of contact. As will be remembered, the segments of the string are separated from each other by points of apparent rest called nodes. Of course, these nodes are not actually at rest, but the amplitude of their motion is greatly restricted by reason of the opposed forces pulling from each

56 Modern Piano Tuning.

side upon them. If now the exciting blow is struck exactly on one of the nodes, the vibration of the shorter of the two segments into which that node divides the string, and equally the vibrations of all multiples thereof, are blotted out. Thus, if we wish to eliminate the 7th partial, we must strike on the 7th node, that is to say at exactly Vi of the string's speaking length. It is obvious that since the first six partials are simply components of the common chord of the fundamental or 1st partial, and the 8th is triple octave thereto, the elimination of the 7th will produce a perfectly harmonious flow of partials and in consequence a full round mellow tone. Experience confirms this deduction, although the exigencies of piano build- ing usually compel a striking distance, as it is called, positioned at Vs or even higher for the greater number of the strings, and running pro- gressively higher in the upper treble till it some- times reaches /44 at the extreme C7. The influence of contact point position is thus clearly shown, for if any of the very high strings be purposely struck at lower points than the hammers are fixed to strike them, it will be found that their tone is less bright, more mellow and even feebler. The last quality is due to the fact that the prime

On the Vibration of a Piano String. 57

or 1st partial of these short stiff strings is not sufficiently powerful of itself and needs the back- ing, as one may say, of many partials to give it consistency and ^^ring." It might be remem- bered incidentally that in the two highest octaves of the piano the progressively higher contact points of the hammers on the strings introduce series of partials running from the first ten to the first twenty. But the longer and more natu- rally powerful strings are struck at about Vs of their distance and would often be bettor off if struck at Yt.

Material. The properties of the material from which the string is made are also of importance in considering the precise nature of the mixture of partials which any given example may show. The stiffer a string is, other things being equal, the more rapid and complex will be the reflections of wave-motion and the consequent formation of ven- tral segments. By stiffness I do not mean thick- ness; for of course the thicker the string the less intense will be the wave-reflections and the fewer the high partials produced. But the piano is peculiar in that the tension of its strings does not vary largely from one end to the other, whilst the thickness does indeed differ very largely in

58 Modern Piano Tuning.

proportion, since even in the understrung part of the scale the difference betwen the extreme treble and the first above the overstrung will usually be something like the difference between 5 and 8. So it follows that the upper treble strings are very- much stiffer than those in the lower regions, in proportion to their length. Of course, the length factor enters into the complex here too, for the higher strings are shorter, and so again stiffer, for any given stretching force. / / Wire density. In the circumstances it would

seem, after one has tested various pianos of vari- ous grades, that the idea of intensely hard wire is most distinctly a wrong idea; at least if we are trying to get round full tone and not hard glitter. The very hard wire is no longer so generally de- manded, and piano makers are begining to require a string of softer steel which shall tend to produce, under the lowered tension conditions thus made necessary, vibrational mixtures involving fewer ventral segments, the upper of which with their consequent partials shall be less prominent.

String Tension. A softer wire cannot with- stand excessive tensions. But we can easily see that high tension means stiffness, and one only has to listen critically to the tone of most pianos

On the Vibration of a Piano String. 59

to realize that their strings do not err on the side of resiliency. They are usually too stiff as it is, and although the craze for clang and noise seems to be dying out — for which we should be thank- ful— still, there is much to be done yet. The piano of the future, let us hope, will be a low tension piano, equipped with softer wire and with a ham- mer shaped and positioned to kill the 7th harmonic and all its multiples ; a piano which will have few partials in its tone above the seventh and which in consequence will evoke sounds, full, mellow and sustained in quality.^

Voicing. In Chapter IX of this book, I make use of the material here set forth in order to show the practical application of Acoustical science to the work of tone-regulating or voicing pianos by manipulation of the hammer felt.

Simultaneous String Vibrations. We shall now have to face the last and in some ways the most fascinating of all the subjects which we shall con- front in the course of our examination into the vibrations of the piano string. So far we have

1 other piano string characteristics: For some special cases exhibited by piano strings under practical conditions, the reader may consult Chapter VII. Piano Ibass strings: The special cases exhibited by the covered strings for the bass tones, are dis- cussed in Chapter VII.

60 Modern Piano Tuning.

confined our thought to individual strings sound- ing alone. We now have to consider the very beautiful and important phenomena arising from the sounding of two tones simultaneously. The inquiry is of the utmost importance in the higher analysis of piano tuning.

Beats. The piano serves us again to good pur- pose in examining the behavior of simultaneously sounded tones. Let us damp off one string in a triple unison on the piano. (All strings of the modern piano above the overstrung section are strung with three strings to the note.) This will leave two strings vibrating. If the piano has not been tuned very recently, it is almost certain that when we listen carefully to the sounding of these two strings we shall hear a sort of sound that can only be described as discontinuous and **wavy." In order to make sure, suppose we choose the strings corresponding to Cg = 258.65 (middle C at international pitch). Let us damp one string of the triple and then slightly turn the pin of one of the others so as definitely to put it out of tune with its fellow. A very slight turn, just enough to feel the string give, will be suffi- cient. Now, take a tuning fork sounding exactly the same international pitch Q^= 258.65. Sound

On the Vibration of a Piano String. 61

it, and listen carefully. You will hear a clear con- tinuous tone, which persists without deviation or fluctuation. Put aside the tuning fork, strike the piano key, and listen. By contrast you hear a con- fused medley of sound, in which the fundamental pitch is discernible, but is surrounded by, and buried in, a mass of wavy, fluctuating, rising-and- falling sounds of a peculiar character, which are unmistakable when once heard. The peculiar character of these sounds is their rise-and-fall ef- fect. The tone swells out far beyond its normal intensity and then dies away. This wave-like sound rises and falls at definite intervals, and it will be seen that the further apart in pitch the two strings are, the more of these rise-and-fall periods there will be in a given time. If now we begin to turn the tuning-pin backward, so as to bring the disturbed string again to its original equality of pitch with the other, we shall find that the wavy sounds gradually become slower and slower, until at length they disappear, and only the pure continuous tuning-fork tone remains; showing that the two strings have now been brought into perfect accord. Let us see what causes this interesting phenomjenon. Condensation and Rarefaction. We must go

Figure 10.

62

On the Vibration of a Piano String. 63

back for a few moments to some earlier considera- tions. A sound-wave is an oscillation to and fro. When a tuning fork prong vibrates, the first half of the vibration is when the prong moves away from its rest position and pushes the air in front of it against the surrounding air. This pai-t of the vibration has the effect of compressing the air on that side, whilst on the other side the air moves forward to fill up the vacuum left by the moving away of the prong and thus is rarefied or thinned. Consider a vibrating pendulum (Fig. 10), and think of it as if it were a slow moving tuning fork. As the pendulum moves in one direction it con- denses or compresses the air in front of it, and then as it moves back that same air is again rare- fied or thinned out to its original density ; for air is elastic and rebounds. Thus each complete vibra- tion of tuning-fork, string, or pendulum, no mat- ter how slow or rapid, produces a condensation followed by a rarefaction of the surrounding air. Wave-length. The space or distance between one condensation and the next, or between one

Figure 11.

64 Modern Piano Tuning.

rarefaction and the next, is called the wave-length. The more of these pulses there are in a second or other unit of time, the shorter the length of each. Sound travels at the rate of 1100 feet per second, roughly speaking — the wave-length of a tone of 100 vibrations per second therefore is ^^^%oo or 11 feet. Figure 11 illustrates this point.

Phase. Thus we see that a sound-wave propa- gated through the air consists of a series of these oscillations of rarefaction and condensation.

Figure 12.

Now suppose that you start two such wave systems simultaneously from two strings perfectly in tune. Start them exactly at the same time so that the condensations begin together. A good example is the striking of two strings at once on the piano. The two run exactly together, condensation with condensation and rarefaction with rarefaction, as is shown by Figure 12, and are said to be in the same phase. Difference of Phase. Now suppose we can ar-

On the Vibration of a Piano String. 65

range to start one string vibrating just half a complete vibration beliind the other. Then con- densation of No. 2 begins with the first rarefac-

FlGURE 13.

tion of No. 1 and we have the state of affairs pictured in Figure 13. Such a condition is called difference of phase.

Difference of one vibration. Suppose two piano strings, one of which gives just one vibration less per second than the other. Now, when these two strings are sounded simultaneously, it follows that at the end of a whole second one will be exactly one vibration behind the other. Likewise at the end of half a second one will be half a vibration behind the other ; or in other words at the end of half a second, or right in the middle of one sec- ond's complete series of vibrations, the two will be in different phases, while at the end of a whole second they will have regained identity of phase ; will be in the same phase together again. Sup- pose now we lay out on paper two wave systems, whose frequencies shall be in the ratio 8 : 9, for the

Modern Piano Tuning.

^<

C

sake of simplicity. Let us also show, by a third wave-curve, the result of the simultaneous activi- ties of the two waves. In order to avoid a complex drawing I show just eight vibrations of the one and nine of the other. These will con- sequently begin and end together.

Resonance and Interference. Now as soon as we examine these superimposed curves, we see that at the second complete vibration they are distinctly out of step with each other and by the time one has made four complete vibrations they are in definitely opposite phase. From this point onwards the difference subsides until at the eighth vibration of the one and the ninth of the other, the phase is again the same for both.

Now, it will at once be seen that when the two waves start, two condensations come together and so we have one condensation on top of the other, which of course

66

On the Vibration of a Piano String. 67

means an increase in amplitude of the combined sound. Hence at the beginning of the waves the sound of the combined tones will be increased over the sound of either of them alone. We have a condition of resonance, as it is called.

On the other hand, when the middle of the curve has been reached we see that the condensation of one meets the rarefaction of the other exactly, so that at this point the one wave blots out the other and produces a perfect interference as it is called, cancelling the sound altogether.

Hence we have the rise and fall of sound which we heard so clearly in the two piano strings men- tioned above.^ This rise and fall is very distinct and in the present case would occur at each 8-to-9 period; in other words, if the two waves were vibrating at 80 and 90 vibrations per second re- spectively, there would be heard 10 beats per sec- ,ond between them when sounded together.

Frequency of Beats. Beats therefore arise be- tween sounds nominally in unison but actually slightly out of tune with each other. The number of beats in a given time is equal to the difference between the frequencies of the generating tones.

Coincident Partials. Beats arise only between

1 See pages 60 and 61.

68 Modern Piano Tuning.

unisons. When heard in such intervals as the Octave, Fifth, Fourth, Third or others, this is because partial tones which may be common to both are thrown out of tune slightly ; and the beats arise between them. For instance, the beats in an octave which is somewhat out of tune arise between the prime of the upper tone and the second of the lower; which are the same. Ex- ample: C, = CA and C2 = 128. Prime of the higher is 128. Second of the lower is 128.^ These are therefore coincident, and if the strings which produce the primes are not in accord, the coinci- dent partials will generate beats as above. The same is true in the interval of a perfect Fifth where the coincident partials are the 2nd of the higher and the 3rd of the lower. Please observe that the coincident partials always bear the same num- bers as express the ratio of the fundamentals. Thus octave ratio = 1:2 and coincident partials are 2 and 1. Fifth ratio ==2:3 and coincident partials are 3 and 2. Fourth ratio = 3:4, and co- incident partials are 4 and 3; and so on for all other intervals. For instance: Suppose one tone = 200 and another = 301. The interval is a Fifth, slightly out of tune, as the higher should be

1 See supra, p. 53.

On the Vibration of a Piano String. 69

300. Coincident partials are 3d lower and 2d higher. 200 X 3 = 600 ; 301 X 2 = 602. 602 — 600 ^ 2 ^ number of beats per second in this out-of-tune Fifth.

Use of Beats. From what I have said, it be- comes plain that the tuner will find beats very use- ful and must devote himself to practicing the art of hearing them and counting them. For it is evi- dent that if the number of beats between any two coincident partials is equal to tlie difference be- tween the frequencies thereof, then if we calculate the exact required frequency of each of the two members of an interval and from this calculate the frequency of their lowest coincident partials, w^e can easily and at once take the difference be- tween the two latter, and whatever this differ- ence be, that number of beats per second will be heard between them when sounded together. Therefore if we tune the two members of the in- terval so that we hear just that number of beats per second between them, we have tuned cor- rectly. It then remains only to calculate these true values for the different tones of the piano and thus to establish proper beat-rates everywhere.

Miller's Researches. There is nothing unusual in all this really, for of course all tuners tune by

70 Modern Piano Tuning.

counting beats, whether they call the process by this name or another. The point I am making here is that this process is the proper and natural process and that it is capable of being established mathematically, as has been done by Mr. J. C. Miller of Lincoln, Neb.; whose researches I am happy to be able to make use of in this book, as will be seen later.

We have now discussed to such an extent as is necessary for our present purposes the behavior of piano strings in vibration ; and have discovered that this discussion, if properly understood, is found to give us all needed assistance in both tun- ing and voicing, provided we can calculate the necessary frequencies of the tones required on the piano. We already know ^ that the piano does not permit pure tuning of the diatonic scale but that a system of compromise must be adopted to ac- commodate the inequalities of the diatonic scale to the unyielding 12 keys of the piano's octave. The system of Temperament used for this pur- pose, called the Equal Temperament, is now so firmly ingrained in practice that it is in fact the real basis of all modern music; rather than the diatonic scale, which indeed is now little more

1 Supra, Chap. I.

On the Vibration of a Piano String. 71

than an artificial abstraction. Of this, however, I shall speak in the next chapter.

If the present chapter has seemed at all involved this is only because I have had to treat an in- volved subject in simple language and small space. Still, all I have said here has been necessary and forms part of the argument which I am develop- ing as to a system of piano tuning and tone-regu- lating; based on science and not on guesses or rule-of-thumb — and a good deal easier than if it were so based.

For, indeed, among all the ridiculous supersti- tions of the human mind, none is either commoner or more absurd than that which covers with con- tempt the efforts of pioneers to formulate and apply scientific method. In truth, to do things scientifically is always to do them in the easiest as well as the best, way; and your ^'practical man," untainted with one touch of theory, wastes time and energy in equal proportion.

Chapter III.

TEMPERAMENT.

We have reached the central position in the science of tuning. What has gone before has been enough to show that one cannot obtain a series of pure diatonic scales, in the quantity required for the performance of music, with a key-board com- prising only twelve keys to the octave. The par- ticular method adopted in Chapter I for the pur- pose of showing the truth of this assertion might of course be matched by a dozen others; without altering the facts in the least. For example, I might have pointed out that an ascending series of perfectly tuned perfect Fifths, although nominally equal to seven Octaves, yet actually exceeds them. I might have shown that three major Thirds should be equal to an Octave, if tuned pure one above the other ; but that in fact they fall considerably short thereof. There are many other possible illustra- tions; but I have already shown, in the simplest

72

Temperament. 73

manner, that some form of compromise is needed if pianos are to be tuned so as to make the per- formance of music in all tonalities tolerable de- spite the defective and inadequate 12-to-the-octave key-board.

The word Temperament is generically used to indicate any one of the many such systems that have been, at one time or another, proposed and used. It must be remembered that the present type of key-board dates certainly from the 14th century and has scarcely undergone any change in details — positively none in essentials — during all that time.^ This is an amazing commentary on the slowness of the human mind and its hatred of change. It is a fact that the width of an octave, even, has remained the same for certainly three hundred years. And the same slowness of de- velopment is true in other details.^

Influence of the key-hoard. The truth implied

1 A terra-cotta model showing a rudimentary form of key- board used with an Hydraulikon or water-organ, has been found in the ruins of Carthage, and is assigned to the second century A. D. Cf . A. J. Hipkins' "Introduction to the Key-board Instru- ment Collection," Metropolitan Museum of Art, New York.

2 The great organ at Halberstadt, Germany, built in 1361 by the priest Nicholas Faber, had a complete chromatic key-board, but with very wide keys. However, sixteenth century clavichords are preserved showing key-boards essentially identical with that of the modern piano in width and even in moimting.

74 Modern Piano Tuning.

in Chapter I may now be realized completely : that the key-board has always exercised a distinctly enslaving influence upon the development of music. If we were not chained to the 12-note key- board by the tradition of music teaching and of piano making, we should soon have a substitute, as easily taught to the hand, whereby at least the grosser imperfections of any temperament sys- tem might be avoided. But to hope this is to hope too much.

Meaning of Temperament. Actually, the word Temperament means ' ' tuning ' ' ; nothing else. Its derivations from the Italian and thence from the Latin, show this clearly. To ''temper" sounds is to tune them. And this fact indicates that the necessity for a compromise from purity was rec- ognized very early and that just intonation has never been even near accomplishment in ordinary practice. In fact the system of Temperament now in use is probably the best that has yet been contrived, although it has had one rival whose claims are not to be despised.

Equal Temperament. The twelve keys within the octave must, of course, represent amongst themselves the various degrees or steps of rela- tionship existing within that interval. Seeing

Temperament. 75

that we cannot gain purity of ratio with only twelve keys, it follows that we must divide up the octave in some way that will admit, as adequately as may be, of performing required music in a toler- able manner. Equal Temperament is the name given to a system of dividing up the octave into twelve equal parts. This being the case and the pitch proportion of the octave interval being 1 : 2, it follows that the proportion from semitone to semitone in equal temperament is 1 r^y or 1 : 1.0594631, correct to seven places of decimals. This ratio is the ratio of the equal semitone, upon which the system is based.

The Equal Tempered Scale. This being so, we have only to select some standard of pitch for some one tone and calculate up and down there- from by the simple process of multiplying or divid- ing, semitone by semitone, by the factor 1.0594631.

The octave of course remains the one interval which retains its purity. This is so, because we must have a system of some sort and the octave provides a foundation therefor. Hence the octave remains pure, and so if we once calculate the equal tempered pitch of the 12 semitones in one octave we can obtain that of any one of the tones situated in any other octave by multiplying by 2

76 Modern Piano Tuning.

for eacli octave of distance upwards or dividing by 2 for each octave of distance downwards.

Thus we may say that Equal Temperament is a system in which the octave interval is tuned pure and all other intervals are tuned in such a way as to produce a tone-series of 12 equal parts within each octave.

International Pitch. The nominal standard now recognized for the basis of pitch in the United States is A3 = 435. This is the same as the French Normal Diapason, from which indeed it is taken. Assuming this as our standard, we have the following frequencies for the A throughout the compass of the piano, beginning at the lowest :

A-,	- 27.1875
A	— 54.375
Ai	- 108.75
A.	= 217.5
A3	— 435
A,	- 870
A5	-1740
A«	-3480

The piano's scale in equal temperament. With the above figures as our standard of measurement, and multiplying for each semitone upwards by the

Temperament. 77

equal semitone ratio, we get the table on the next page, showing the complete range of frequencies for the entire 88 notes of the piano.^

Object of the Table. My object in setting forth this Table is not to confuse the student but to enable him to see how a system of tuning in Equal Temperament may easily be worked out; one far simpler and considerably more likely to lead to correct practice than any other based on guess. The Table is the preliminary essential in the argu- ment now to be set forth.

By examining the Table we observe that if any column be taken, the figures from top to bottom thereof represent the progression of frequencies in the sounds of an ascending octave. All columns to right of any such column are ascending octaves and all columns to the left are descending octaves. From column to column we may proceed by either multiplying or dividing by 2 at each column. The octave interval is pure and the others are

1 In reality the names used for the tones in the equal tem- pered scale are incorrect, since they are the same as those of the pure diatonic scale. For purposes of convenience we continue to say C, B flat, G sharp and so on, whereas it would be more ac- curate simply to number the septave C to B as 1 to 12. How- ever, seeing the musical notation still sticks to the old key-sys- tem, though this no longer means anything in point of char- acter, the old names are respected in the table, the sharps and flats being noted as coinciding.

(^

93 o	.>* • 00 ; • • eo
4) > o	o • • .(MOO'*<M-<*<0000'*i» • ■ ■ ■00(MCOO«0(>aOOOCOODO ■
> o	■ ■ 'Tiicoi— locor-icoocgococa ; ; ; ;oo^(Mco<ro-*io<»t~ooffli |
o O	... o • • • • CO O U5 rH CO CD in O O O CO ■*. • ■ ■ ■ t--^ cc o in ^' o i-< in i-H o .-H «o | . . . 1— 1 Tf CO »-i lO Oi «> 1^ (M 1-^ <M 1^ . ininincoocDt-t^QOcooiO
■D o o	• • • in o • ■ •coooiinoscot^ininC'CCOJ • ■ ■ 'co-^ot^ininint-^oinooo ' ■ ■ 'inir^CSOtN-^COOCi— cfOtoQO .
o	. . . OJ o • ■■ •coOr-;t--oieqoqi>-c<]in'^i-; ■ ; ■ ; ffi t-^ in CO cj oi (m' cc in i~-^ o 'i^ " (McoTjHincDt^oooiO^cc-* ■ i-lrHr-(i-HnHr-<>-H,-l<MiM(M(M
o o	■ • -cooininin m m in- • • • .-H lo iq GO Tj^ CO Tj; CO «5 1-- (>i o • ; ; ■ •>*' CO oj CO F-H !» ^' (ffl oq CO in (m' ; COCOt^t-COOOOCiOO^tN i-H 1— 1 .— 1 1— 1
c Octave	• • • CO in t-- (M t^ in c-1 t- • • • O l?I C<1 ■^_ t--; >-; I--; M; CO CO CO o • ■ ■ ■ c^i -t CD co" o co' in" 00 I-H -+ t-I »-i ; cocococoTtiT»<-<*'<tiin»nincD
Sub Octave	00 r-HCoin
oi oi CO
(V B

78

Temperament.

79

worked out on the equal semitone system as ex- plained.

Comparison of Intonations. Before undertak- ing to show how tuning in Equal Temperament may most easily be performed, I shall give here a comparison, for the student's benefit, of three pure major diatonic scales built on two of the tempered tone degrees taken from Table I.^ The first scale is the tempered scale, the second is a pure diatonic

TABLE II. COMPARATIVE FREQUENCES OP TEMPERED SCALE

AND THREE DIATONIC MAJOR SCALES TAKEN ON

1ST, 2ND, AND 3RD CHROMATIC TEMPERED

DEGREES OP TEMPERED SCALE

Pure Diatonic Pure Diatonic Pure Diatonic C Scale D Scale Dp Scale

(Major) (Major) (Major)

Cj ....258.65

Dj, .... 290.2 Dp .... 274.00

D 290.98

E 326.47 EJJ .... 308.25

E 323.3

F 344.8 FJf 362.75 F 342.5

G 386.93 G> 365.33

G 387.97

A 435.3 Ap ....411.00

A 431.08

B 483.60 Bl? 456.66

B 484.97

O3 517.3 C 513.75

CJf., ...544.12 Dl> 548.00

Equal Tempered
	C Scale
( Chromatic )
c,	. > •	..258.65
CJ-	-Dl>	..274.00
D		..290.2
DS-	-E^	..307.5
E		..325.9
F		. .345 3
F#-	-GJJ	..365.7
G		..387.5
G#-	-AP	..410.5
A		..435.
A#-	-Bi>	..460.8
B		..488.2
0,		. .517.3
c#-	-Dl>	..548.
D		..580.

D,

.580.

1 The minor scale has not been considered because its difference from the major is merely in the detail of intervals. The argu- ments already made apply with equal force to the minor scale. See supra. Chap. I,

80 Modern Piano Tuning.

major scale built on the same Cg = 258.65, tlie sec- ond in a pure major diatonic scale built on the tem- pered major second to C (D), and the third a pure diatonic major built on the tempered D flat. The pure diatonic scales are worked out from each on the basis of the ratios of the diatonic scale major {supra Chapter I) ; and the object of the comparison is simply to show what effect the Equal Temperament has on purity of intonation.

Advantages of Equal Temperament. These tables show clearly some of the peculiar defects of the Equal Temperament; but they show also some of its peculiar advantages. For it will be seen that at the cost of some perceptible disso- nances in certain intervals — dissonances which we shall shortly calculate definitely — we gain the abil- ity to perform music in all tonalities, by aid of the traditional 12-note key-board.

Disadvantages of Equal Temperament. At the same time we must not lose sight of the fact that in reality the Equal Temperament is a comprom- ise, and a loose compromise, with fact. If it were not for the organ and piano, the imperfections of Equal Temperament would be more easily per- ceived; but the dynamic powers and immense

Temperament. 81

harmonic resources of these two instruments have endeared them to musicians and have concealed the roughness of their intonation. No one who has read the previous chapter and understands how to listen for beats, however, can long endure the intonation of the organ on such intervals as minor thirds. The sustained tones of that instru- ment bring out beats very clearly and produce a generally distressing effect for delicate ears. Of course, the truth is that most of us are so used to tempered intonation that we recognize nothing else and know of no other possibility. Yet the fact remains that whoever has heard one of the few experimental key-board instruments that have been constructed to play in pure intonation has been entranced with the sweetness of music thus played. It is far more beautiful than tempered intonation and in fact seems to impart to the music of these instruments a new sweetness and concord. So long, of course, as the manufacture of pianos and organs is stressed rather on its industrial than on its artistic side we shall probably have to remain content with Equal Temperament. But it might as well be observed that if the piano and organ were out of the way, music throughout the

82 Modern Piano Tuning.

world would be on some basis of tuning other than Equal Temperament within ten years. ^

Meanwhile we must be content to tune in Equal Temperament as well as we can, knowing that when such work is well done it is very satisfactory and serves well the requirements of modern music and modern musicians.

Meantone Temperament. Before going on to consider the method of tuning in Equal Tempera- ment, however, I should like to mention the imme- diate predecessor of the Equal Temperament ; the famous Meantone Temperament, which flourished from the 16th to the early part of the 19th century and may be occasionally found to-day on organs in obscure European villages. This system con- sists in tuning a circle of fifths equally flat, in such a way as to leave all the thirds major nearly pure. In order, however, to be used for all re- quired keys, it is necessary to have extra key- levers, for the flats and sharps of adjacent tones are not identical. For perfect performance in all tonalities, not less than 27 tones to the octave are

iThe reader who doubts this miglit consult Ellis (App. 20 to Helmholtz), Helmholtz, chapter 16, Perronet Tliompson, "Theory and Practice of Just Intonation," and Zahm, "Sound and Music." My own book, "Tlioory and Practice of Pianoforte Buildin<^," con- tains a close analysis of the requirements of Just Intonation,

Temperament. 83

required, but tlie greater number of tonalities can be used with 16 keys to the octave ; the additional tones being for D flat, E flat, A flat and B flat. The ordinary 12-tone key-board would give, of course, starting from C, only the circle of Fifths, which when transposed to the same octave result in the following scale :

C, C#, D, D#, E, F, F#, G, Gj, A, Ajf, B.

Unfortunately, in this temperament, C# will not do for D flat, D# for E flat, G# for A flat or A# for B flat. These tones of course have to be incor- porated somehow and in some 18th century organs were built into the manual by dividing up some of the black keys, which were cut across the mid- dle with the back half slightly raised above the front. The mean tone system gives a ''sweet" and harmonious effect for nearly all keys, with 16 tones to the octave, although of course this num- ber still lacks 11 tones to make it quite adequate. However, even with 12 tones to the octave, an ex- periment in meantone temperament can be tried, and will sound veiy attractive so long as one keeps within the range of keys allowable. To make the best of the key-board we have, the following method may be tried. Start with C and tune the

84 Modem Piano Tuning.

major third C — E perfect. Then tune the fifths from C round to E by Fifths and Octaves, equally flat, testing until the right degree of flatness is ob- tained. All other notes can be had by tuning pure major Thirds and pure Octaves. By this system it is possible to play in the keys of B flat, F, C, D and A major, and G, D, and A minor. The reason, of course, is as stated below. This is a very use- ful experiment and if tried out carefully will en- able the student to play old music in the tuning for which it was intended; an experience sometimes most illuminating and delightful.^

The ''Beat" System. I have mentioned these things because I am anxious to have the student understand that the Equal Temperament is not the only possible system of tuning. But to get now definitely to the method of tuning in Equal Temperament, which is the system which the tuner to-day uses universally, let us see what is the nature of our problem.

The Table of frequencies (Table I) suggests the method we shall use. We know ^ that beats afford

1 The student might consult the extremely useful and interest- ing article, "Temperament," by James Lecky, in Grove's "Diction- ary of Music and Musicians."

2 Supra, Chapter II.

Temperament. 85

us a simple and accurate way of judging the devia- tion from consonance of one tone sounded with an- other. Since we cannot trust the unaided ear to tune successively a series of equal tempered semi- tones, we make use of the method of comparing one tone with another. Thus we have only to as- certain the number of beats that are produced by the members of various intervals in equal tem- perament, beating against each other, and then to tune these intervals by counting their beats.

Beats arise between coincident partial tones and therefore if we lay out a series of intervals from some given standard tone and calculate the co- incident partial tones in each, we can by subtrac- tion find out how many beats there are heard when that interval is rightly tempered.^ Experience shows that it is easiest to tune by Octaves, Fifths and Fourths; by Fifths and Fourths for the Oc- tave of tones, usually Fg — F3, chosen for the ** bearings" or foundation work and by Octaves up and down thereafter. The other intervals in- volved are best used for testing the correctness of the work as it proceeds. Such testing is best

1 Compare Chapter II, "Coincident Partials," to find what par- tials coincide in any interval.

86 Modern Piano Tuning.

done by means of major and minor Thirds and major and minor Sixths, whose rates of beating in equal temperament the tuner must therefore know.

Beats in Equal Tempered Intervals. The fol- lowing Table (Table III) gives the number of beats per second in the ascending minor Thirds, major Thirds, Fourths, Fifths, minor Sixths and major Sixths from each degree of the equal tempered scale between Co and C^ inclusive. The rates are, for purposes of convenience, made correct only within .5 vibrations per second. In other words, where an accurate calculation would show any beat-rate as some whole number plus a decimal greater than .5, the rate has been made the next whole nuinber. For instance, 19.73 is counted as 20; while the same course has been adopted for rates where the decimal correctly is less than .5. For instance, 9.31 is made to read 9.5. On this plan the error may be less than .1 or more than .4 vibrations per second. Inasmuch, however, as the tuner will find his powers extended to the utmost in estimating the beat-rates of Fourths and Fifths at the figures given, and with this relatively large error, it has been thought better to adopt this course.

For suggestions as to counting beats and other

o •

03 ;^

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lO -*

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inincdcocdt-^t-^xodcJooO'-<'-Hc4co-^Tt<ioco

is

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l«

lO

IC

lO

lo m

lO

t^t— OOOOOi0500'-i(N(MCO-^i.'5COt~CCCii— iF-HcaCO

COIC f-li-lr-l^i— l,-l>-l,-l^r-Hrtl-lCg(NC<l<M

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d a)

S

coOl-^^-;0<qco^-(^alftTt;r^coo(^5»nc5eol>;lq^qOoq(^^eo_ o t~^ lo CO e4 (N N CO lo t^ o T)< oo TjJ o (--• lO lO lo 1--^ o ic o oc 1^-

(NCOT)HmcOt^COOOi— <C0Tt<l0t--C;O(M'^C0Q0i— iCOCCCO'-< I— l,-(rt,— (^,-l>-ii— i(N(MlM(5qCNC<l(MCOCOeOeOCO'*"^Tt"'*lO

o

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43 ^ u bo

a

o • xv ; xi. . :x^ .x^ . x\ : .x\ i^^ . .j=^ .x^ --^

O -Q -H • -O •<) -PQ . -Q -W ■ -O ■< •«

. ■ : : I ■ • I ■ I : : I • I :

87

88 Modern Piano Tuning.

practical matters of the same sort the following chapters should be consulted.^

Use of the Table. I do not propose that the tuner shall try to count accurately the beats per second enumerated above; at least immediately. But the special use of the Table is to provide a model which will indicate closely enough the exact amount of impurity in each interval as required for equal temperament. This impurity is meas- ured by the beat-rate instead of by first showing what the ratios of impurity should be and then proposing a rough approximation thereto to be measured by tuning ** about so flat" or "about so sharp." By giving definite beat-rates I make it possible, as the next chapter will show, to tune with unusual accuracy after a reasonable amount of practice.

The immediate point is that the tuner should

1 The present table is founded on the comprehensive and ac- curate work of J. C. Miller of Lincoln, Neb., U. S. A., an eminent exponent of the noble art of tuning and a worker in science whose unselfish labors for the benefit of his fellows can never be too much admired. Calculated correct to six places of decimals and with some supplementary matter, the Miller tables were pub- lished in The Tuner's Magazine (Cincinnati), for December, 1914. A less elaborate edition was published correct to two places of decimals in the Music Trade Review (New York) during 1911, and in the London Music Trades Remew (London), 1914. Those who desire to obtain the complete figures may therefore have them by consulting the publications mentioned.

Temperament. 89

know bow many beats per second eacb interval in tbe equal temperament scbeme really involves, at standard pitcb. Knowing tbis be knows wbat be ougbt to do ; and if be can learn wbat be ougbt to do, tban be can sooner learn bow nearly to attain in practice to tbat ideal.

Wide and Narroiv Intervals. It only remains to note wbicb intervals are to be widened and wbicb narrowed in equal temperament tuning. Tbe facts are simple and easily understood. Tbey may be stated as follows :

An ascending series of twelve Fiftbs nominally coincides witb an ascending series of seven Oc- taves from tbe same notes. Actually tbe Fifth series comes out sbarper tban tbe Octaves ; in tbe ratio 531441 : 524288. Hence equal tempered Fiftbs must be narrowed ; tbe amount of flatting in eacb case being determined for tbe tuner by count- ing tbe beats as set fortb in Table III.

An ascending series of twelve Fourtbs nominally coincides witb an ascending series of five Octaves from tbe same note. Actually tbe Fourtb series comes out narrower tban tbe Octaves in tbe ratio 144 : 192. Hence, equal tempered Fourtbs must be widened, tbe amount in eacb case being deter- mined for tbe tuner by counting tbe beats.

90 Modern Piano Tuning.

Although we tune by Fourths and Fifths prefer- ably, it is necessary to understand also the charac- teristics of the other intervals. Thus :

Three ascending major Thirds nominally coin- cide with an Octave but actually are short in the ratio 125 : 128. Hence the major Thirds must be widened in equal temperament, the amount thereof in each case being determined as in Table III.

Four ascending minor Thirds nominally coin- cide with one Octave but actually exceed it in the ratio 1296 : 1250. Therefore they must be nar- rowed in each case, the amount thereof being de- termined for the tuner by counting the beats as set forth in Table III.

In the same way we find that in Equal Tempera- ment the major Sixths are all wide and the minor Sixths narrow.

Increase in heat-rate. In the nature of the case, the beats taking their origin from coincident par- tials, and the ratio of pitch from octave to octave being 1:2, it follows that the number of beats in any interval doubles at each octave ascending and halves at each octave descending. By suitable multiplication and division therefore, the beat-rate in any interval in any octave of the piano may be obtained from Table III. It is worth while pans-

c .	..F	.11
c,.	..F,	.22
c	..F,	.45
Ca.	..Fa	.90
0,.	..F,	1.80
a.	..F,	3.60
CV	..Fe	7.20

Temperament. 91

ing here to note that the higher octaves comprise intervals in which the beats are very rapid and conversely the lower octaves have very slow beat- ing intervals. Thus the ascending Fifth C — ¥ beats as follows, from the lowest C upwards :

V. p. s. .1125 or a little more than 1 in 10 seconds, or nearly 5 times in 20 seconds, or nearly 5 times in 10 seconds, or 9 times in 10 seconds.

or nearly 2 per second (18 times in 10 seconds), or nearly 4 times per second (36 in 10 seconds). or about 7 times per second (72 in 10 seconds).

The Fourths and Fifths Circle. My preferred method of tuning is by a circle of Fourths and Fifths. By the method noted on pp. 108-109, Fourths and Fifths are alternated in such a way that after the first descending Fifth has been tuned, every note to he tuned is flatted, whether it belong to a Fourth or Fifth. This is made pos- sible by taking all the Fourths descending and all the Fifths ascending, whereby Fourths can be expanded and Fifths contracted in a continuous process of flatting each note in turn; as will at once be seen by glancing at the table mentioned above. ^

Practical suggestions for tuning this circle ac- cording to the beat-rate system will be found in

1 See pp. 108-109. '

92 Modern Piano Tuning.

the next chapter. The tones are taken in the octave F2 — F3, around middle C, and the re- mainder are tuned by octaves up and down. The circle is known among tuners as "the bearings."

A note on the Scientific Method. This con- cludes all that it is necessary to say about the acoustical foundation of the system universally employed throughout the Occident for the tuning of musical instruments. The practical side of the art is treated in the folloAving chapter. I should not wish to conclude the present remarks, however, without pointing out that I have treated the sub- ject fully but not necessarily obscurely. In point of fact one cannot explain the rationale of Equal Temperament without going into considerable de- tail. If the reader is content to do without the explanations, and to accept Table III together with the following chapter at their face value, he may skip all that I have set forth in previous pages and take my conclusions as stated. But if he chooses any rapid road like this he will find that rapid roads are often slippery, and he will miss the satisfaction which comes from knowing ivhy as well as how.

The attentive reader will discover nothing diffi- cult in the method adopted here ; and the practical

Temperament. 93

tuner may be assured that the system of count- ing beats set forth in this and following chapters is not only quite practical but on the whole easier than any other with which I am at present ac- quainted, and which comes within the limited range of practical requirements. As for accuracy, there is no comparison between this and any sys- tem founded on less exact reasoning.

A Note on Intonations. I should feel that this book has fulfilled its object if it induces some students to take up the study of Intonation in gen- eral; by which I mean the study of the general problem of expressing musical scales in practical form. The Equal Temperament is a good servant but a bad master ; and although the practical piano tuner must, in present conditions, use it as it stands, he cannot divest himself of a certain amount of responsibility in the matter of its gen- eral approval. For after all, tuners could do something to prepare the world for a better sys- tem if they wished to ; and if they knew that im- provement is actually possible. What we need is to realize that the Equal Temperament is a purely artificial system resting upon a consent gained rather on account of the absolute necessity for the piano having it than for any fundamental musical

94 Modern Piano Tuning.

reason. To get sometliing better we need a method which will either (1) allow more strings to the octave or (2) will give us a mechanism capable of making instant changes as required in the pitch of given strings, so that modulation may be as facile as it is now.^

Let it be understood however that Just Intona- tion is an ideal to be striven for; and that tuners should by all means make it their business to ac- quaint themselves with the beauty and sweetness of pure intervals, if only to remind themselves that these really exist.

The scope of the present volume does not permit me to go into detail as to experimental instru- ments that have been made to give Just or nearly Just Intonation, but I wish that every reader would take the trouble to consult the admirable discussion of this most interesting subject to be found in Ellis ' 20th Appendix to the 3rd English edition of Helmholtz. At the end of this book I have ventured to give a short list of works which the student who is desirous of pursuing his ac- coustical and musical studies further, may con- sult to his advantage.

1 Ono such system was worked (by Dr. TTapaman of Cincinnati) some years ago, and one may hope that it may yet be heard of in some commercial practicable form.

Chapter IV.

PRACTICAL TUNING IN EQUAL TEMPERAMENT.

After all tlie discussion that has gone before, we have now to see how we may put our ideas into practice. It is plain from what has been already said that Equal Temperament is a perfectly simple system and one admirably adapted to present ideas and methods in musical composition and the making of musical instruments. Until there comes that change in public taste which demands something finer, we shall have tempered pianos, tempered orchestras and tempered intonation gen- erally; with our ears more and more becoming used to tempered sounds and unused to pure sounds. It is therefore highly important not only that the tuner should be thoroughly capable of do- ing the very best work in temperament that can be done, but also that he should make it his business to realize every day and every hour that the work he is doing is in reality a compromise with truth ;

95

96 Modern Piano Tuning.

necessary no doubt, but a compromise just the same ; and one which exists solely because instru- ments, musicians, and the public alike are un- ready for anything better.

Recognising Just Sounds. In the circumstances and considering the inherent difficulty of temper- ing accurately by estimation of ear, I most ear- nestly advise every student to practice the tuning of pure Fifths, Fourths, and Thirds. Unisons and octaves must be tuned pure anyhow, but most tuners, in common with virtually everybody else, are familiar only with the tempered form of the other intervals and never dream of asking how they sound when purely intoned. But in order to recognize with any sort of accuracy the number of beats per second that a tempered interval is gener- ating, it is absolutely essential that one should be acquainted with the true condition of that inter- val ; familiar with its sound when evoked in purity. Thus I would counsel every reader of this book to form the habit of always tuning all intervals pure and hearing them satisfactorily as such, before at- tempting to temper them ; and also to get into the equally valuable habit of tuning, for his own amusement and satisfaction, series of pure chords on the piano ; which of course can be done, so as to

Practical Tuning in Equal Temperament. 97

be satisfactory for one tonality and even capable of being played in to that extent.

Measure of Purity. The measure of purity is the absence of beats. If intervals whose ratios of frequency are such as to avoid alternation in phase-relations ^ are tuned on any instrument with absolute accuracy, there will be no beats between them. (I here purposely have omitted reference to the ''beat-tones," otherwise known as resultant or combinational tones, which are in reality gen- erated by very rapidly moving beats from high coincident partials in intervals that otherwise would have no beats ; but these beat-tones are true musical sounds and for the tuner's purpose need not be considered in the present discussion ; espe- cially as they are almost imperceptible on the piano. ^) The fact, then, that no beats are heard is a test quite accurate ; and the work of tuning is immensely facilitated if the tuner is able to assure himself when a condition of beatlessness exists, in- stead of always being vague on this point.

1 See supra, Chapter II.

2 Tuners whose ears have attained a sufficient degree of delicacy, and who are scientific enough to wish to go into the matter, may consult Zahm, "Sound and Music," Tyndall "On Sound," Helm- holtz, "Sensations of Tone." Beat-tones, as described best perhaps by Koenig (quoted in Zahm), do of course afford a test of the absolute purity of an interval.

98 Modern Piano Tuning.

Unison Tuning. The acquirement of a culti- vated ear for purity of interval cannot better be begun than by learning to tune unisons correctly. Considering also that tuning of unisons comprises about two-thirds of the work of tuning a piano, since most of the tones are triple, and nearly all the rest double, strung, it will be seen that this im- portant part of the tuner's work deserves the most careful attention. It is fortunate that the unison is the simplest of intervals to comprehend and to appreciate aurally. It is requisite in beginning the study of tuning to ascertain with complete cer- tainty the peculiarities of beats. Two tuning forks tuned in unison and with one then slightly loaded with a minute piece of wax, or other mater- ial, afford a simple and easy experiment in beat production.^ It will be useful to begin by making such an experiment and listening carefully to the beats until the peculiar rise and fall of the sound is so plain that it can never again be mistaken. The opposite experiment, made by removing the load and returning the forks to complete unison, pre- sents the sound of a perfect unison, with complete absence of beats. It is also, by the way, interest-

iCf. Chapter II.

Practical Tuning in Equal Temperament, 99

ing as providing an almost perfect form of simple vibration, without partials.

To tune two piano strings in unison is not a very difficult task if it be gone about rightly. In the first place, it is well to listen carefully to the work of a professional tuner, when the opportunity oc- curs, with the object of noting by ear how he ad- justs his unisons. A little practice will enable one to hear instantly the difference between the beat- ing tone of two strings which are out of tune with each other and the pure continuous beatless sensa- tion of tone developed when the unison is per- fected. Having once obtained some facility in thus cultivating the hearing, the student may ob- tain a tuning hammer and attempt to do some prac- tical adjustment of unisons.

Argument concerning the manipulation of the hammer would at this stage be out of place, I feel, for it would tend rather to confuse than to assist the student.^ Let me then simply say that the tuning hammer is to be placed on a tuning-pin corresponding with one string of a triple unison. Let one string of the triple unison be damped off, leaving two to sound when the piano digital is

1 But see the next chapter.

100 Modern Piano Tuning.

struck. If the piano has not been tuned recently the student will undoubtedly hear beats; and if these are not distinct or frequent enough, let the tension on the string be relaxed by turning the pin slightly toward the left or bass end of the piano. Now, let the reverse operation be under- taken, and the hammer turned back towards the operator, thus gradually tightening up the string again. As this^ is done it will be observed, if the digital is repeatedly struck, that the beats, which at first we may suppose to have been quite fre- quent, become slower and slower, until if the string be rightly tuned, they disappear altogether, leav- ing a pure continuous uninterrupted sound.

Practice. In beginning the work of learning to tune pianos, it is advisable to practice for some time in bursts of from fifteen to thirty minutes each, on simple adjustment of unisons. The manipulation of the hammer, of which I speak at length in the next chapter, now, of course, begins to claim attention ; but for the present it is enough to say that the pin must be turned without wrench- ing its outer end and without pulling downwards on it. It is advisable to rest the arm and keep the handle of the tuning hammer nearly vertical.

It will then be advisable to undertake settling

Practical Tuning in Equal Temperament. 101

the unisons of an entire piano, and not until this has been accomplished is the student fit to take the next step. It will be observed that whereas the middle strings afford a comparatively simple task, those of very low and very high frequencies alike are more and more elusive; and careful practice and listening are required before they can be mas- tered. The measure of purity, be it remembered, is absence of beats; and so long as beats can be heard the unison is not pure.

Octave Tuning. The Octave is the unison transposed. Hence Octave tuning is just as sim- ple as unison tuning, in principle, and almost the same in detail. The beats arising in a two-string unison are between the fundamentals of two sounds whilst in an Octave the beats are between the 1st partial of the higher and the 2nd of the lower ; the Octave proportion being 1 : 2 and the coincident partials therefore being Nos, 2 and 1 respectively.^ Thus throughout the entire middle portion of the piano anyhow, there will be very little difficulty in tuning Octaves, once the student has learned to adjust the unisons accurately, for exactly the same beats are heard and the same blending of the beats into a condition of beatless-

i Cf. Chapter II, pp. 67-68.

102 Modern Piano Tuning.

ness, as the Octave is perfected, until a perfect , Octave is almost as complete a blend of sound as an unison.

Exact Tests. In certain regions of the Piano it is quite impossible to obtain certainty of Octave tuning merely by estimating the extinction of beats between the two members of the interval. In the low bass particularly, the coarseness of the strings and the almost inevitable impurity of their intonation lead to the generation of all kinds of false beats which confuse the student and are a source of annoyance and inaccuracy even to the expert. The student should of course practice tuning Octaves throughout the entire compass, but he will find that in the low bass he must have some test for accuracy better than afforded by aural comparison of the two members of the interval. Fortunately, other tests are available, and each is capable of giving highly correct results.

The Octave comprises a Fourth and a Fifth in succession. A Fourth above Co runs to Fg and a Fifth from that runs from Fo to C3. If now, whilst tuning the octave Co — C:j (say) we sound the Fourth G^—F^ and then the Fifth Fg— C3, and find the beats in the Fourth equal in number to the beats in the Fifth, the Octave is tuned accurately.

Practical Tuning in Equal Temperament. 103

This is so because all Fourths and Fifths in Equal Temperament are distorted slightly so that the co- incident partials are actually a little distorted also. In the present case, the coincident partial is C4, the 4th partial of C2, the 3rd of F2 and the 2nd of C3.

The first octave test therefore is made by ascer- taining the number of beats between the Fourth above the lower sound and comparing these with the beats in the Fifth below the upper sound. If the beats are the same in number, then the oc- tave is perfect. But, if the test is made between lower Fifth and upper Fourth, then the upper Fourth will beat twice as fast as the lower Fifth.

Minor 3rd and major 6th. For the lower re- gions of the piano especially, but useful every- where, is the test of the ascending minor Third and descending major Sixth. If we are tuning, for instance, from Co to C3, we try the minor Third C2 — E flato and note its beat-rate. Then we try the major Sixth E flato — C3 and try its beats. If the beats in the two cases are equal in rate, then the octave is tuned accurately. It will be observed that this is the same idea set forth in the previous paragraph. The E flat is the common tone, and the minor Third and major Sixth are complemen-

104 Modern Piano Tuning.

tary. The coincident partial is the 6th of C2, the 5th of E flats and the 3rd of C^=G^. But if the Sixth be the lower and the Third the upper, com- plementary intervals used, the Third will beat twice as fast as the Sixth.

Beat rates in Thirds and Sixths. It is well to note in passing that as shown in the tables in Chap- ter III (supra), the beats in. Thirds and Sixths, major and minor, are considerably faster than in Fourths and Fifths ; and the student after tuning an octave may profitably examine the complemen- tary Thirds and Sixths within that interval for the purpose of studying their beat rates.

The Tenth test. A Tenth is an Octave plus a major Third. A good octave test is to be found here also by using the major under-Third and Tenth. If tuning C2 to C3, test by the major un- der-Tliird Co— A flatj as compared with the Tenth A flati — C3. The coincident partial is the 5th of A flatj, the 4th of Co and the 2nd of C3,=:C4. This test is useful throughout the entire compass and especially in the lower registers.

Observe that beat-rates, as shown by Table III (Chapter III), vary as the frequencies of their generators, so that the nearer we approach the lower bass, the slower are all the beat-rates. In

Practical Tuning in Equal Temperament. 105

fact, the beat of the Fourths and Fifths cannot be distinguished at all in the lower bass, and we have to depend on Thirds and Sixths, It is very im- portant of course, to know how many beats in a second any given interval should have ; and Table III supplies this requisite.

Counting Beats. The tuner, for practical pur- poses, must count his beats by mere aural estima- tion. The result is never perfectly accurate. It is, however, the only method that can be used in practical work, and the method, therefore, that we must develop to the highest possible degree of excellence. In the first place it will be noted that Table III (Chapter III) gives the beat-rates ac- curate only to within .5 of a beat per second. This is simply because, although the error is mathema- tically large, the human ear, at least with the evanescent tones of the piano, cannot judge any more delicately, and even then needs practice and patience. It will usually be found that for ac- curate work beat-rates of from 2 to 5 per second are most easily counted. Above this is difficult; below it is equally so. Beats of less than 1 per second rate are hard to hear and keep track of. However, the tuner may accustom himself to esti- mating beat-rates by one or two simple methods.

106 Modern Piano Tuning.

■Seconds' Watch. Watchesi are usually provided with seconds' hands and these usually make four ticks per second. By listening carefully for say ten minutes at a time, the student can soon learn to hear accurately a beat-rate of 4 per second. But the precise tick-rate of one's own watch must be accurately determined.

Pendulum Clock. Pendulum clocks of the large sort often have very slow swinging pendulums. These sometimes produce one complete to and fro swing (complete vibration) per second.

Pendulum. Ellis suggests (App. 20 to Helm- holtz) , that the student may make himself a pendulum from a piece of string and a curtain ring, with provision for shortening or lengthening the string readily. Now, for counting during any length of time, as for instance 5 seconds, which is a useful unit of time in counting beats of the piano, we may arrange the length of the string as shown below :

Length of string

from ieginning of Complete vihrationa

vibrating portion {to and fro) to middle of ring

in inches: in 10 seconds in 5 seconds

O'/s 10 5

4% 15 71/2

27%6 6 3

It is important to note the manner of measuring, as above.

Practical Tuning in Equal Temperament. 107

These three beat rates are very important, es- pecially the first and third, in the practical work of tuning.

Other methods may suggest themselves to the student. The perforated rotating disk to be found in every high school physical laboratory can be used also to give series of taps (by holding a card against a circle of the perforations while the disk is rotated), of any required speed.^

Of course the point is that the tuner ought not to trust to guess work but should try to knoiv when he hears a beat-rate what it really is. He can learn to do this; and learn easily. By watching and counting the swings of a pendulum such as that described above he can soon learn to feel the rate of swing; which is an exceedingly valu- able accomplishment and one easy to acquire. Certainly, it is impossible without some such train- ing ever really to do distinguished work in the most important and most difficult branch of tun-

1 The Metronome may likewise be used for the same purpose. The beat-rate required is expressed in terms of beats per minute, then multiplied by 2. The Metronome is set at the resulting figure, and every alternate tick only is counted. Or if a bell is fitted to the Metronome and set to sound on every alternate tick, the re- sult is still better. Thus: 3 beats in 5 seconds = 36 beats per minute. Set Metronome at 72 and make bell sound on alternate ticks. The rate of 3 in 5 seconds will thus be given. The sug- gestion is due to Mr. August Reisig of New Orleans.

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110 Modern Piano Tuning.

ing pianos; that which is called "laying the bear- ings," and which we must now consider.

Laying the Bearings. The scheme already laid out (Chapter III, supra), for tuning the central oc- tave in Fourths and Fifths forms the basis of the explanations now to be made : Reproducing this in type and appending the beat-rate as ascertained for each interval from Table III and the frequency as given in Table I, we get Table IV, pages 108-9.

There are two observations to be made imme- diately. The first is that although the Table shows 36 steps, only 13 notes are actually tuned, and the remaining steps show tests made by in- tervals and chords generated during the progress of the tuning. These test intervals and chords are of the utmost importance, just as important as any other element of the work, inasmuch as they afford a complete measure of the correctness of the tuner's progress. But the actual tuning is confined to the series of Fourths and Fifths shown.

The second is that to count beats at something very close to these rates is not impossible by any means. I counsel the student to realize that any definite method like this has the inestimable ad- vantage of being founded on fact. What is more to the. point, such a method leads to that much

Practical Tuning in Equal Temperament. Ill

closer accuracy which should distinguish masters of the art.

Variations in pitch for practical tuning. The figures given are sufficiently accurate for any pitch between C 258.65 and C 264, which range covers the common variations as found in modern pianos. In strict fact, any rise in pitch above the Inter- national C 258.65 involves a progressive rise in beat-rate. But the amount of increase is too small for practical purposes ; usually at least.

Importance of Accuracy. Of course I can- not too strongly urge the importance of accuracy, of not being content to do things fairly well or even reasonably well; but of insisting to the ut- most upon the possibility of doing ever and ever more accurate, more scientific and more complete, work. Tables III and IV do represent the ut- most in accuracy, probably; and the figures given in these Tables can be attained by those who will practice the Art of Tuning with patient devotion. The Artist will attain them. May every reader of this book earnestly strive for that perfection.

Methods of Using Tests. The test Thirds and Sixths, and the test triads are to be used constantly throughout the work. Observe carefully the rise in beat-rate of the ascending Thirds and Sixths.

112 Modern Piano Tuning.

This rise must be accurately measured. It will be found, of course, that the beat-rate doubles in each octave and the extreme Thirds should show this progress. No better test of the accuracy of the bearings can be found. It is not sufficient to * ' come out right, ' ' by which is meant, to arrive at the last step and find the final octave reasonably clear. It is necessary that every step should be right. This means care, patience, study; which lead to mastery.

Tuning from the Bearings. From the Bearings the tuning proceeds by octaves and unisons ac- cording to the methods laid down and explained in the earlier part of this chapter. The student is again warned to look out that his octave tuning does not begin to slide off from accuracy by the accumulation of imperceptible into intolerable er- rors ; and to this end the constant use of the vari- ous tests already explained, as well as of double octaves and triads, is recommended.

Obtaining required Temperament in intervals. Lastly, let me say again that the only way to tune intervals so that their beat-rates are reasonably accurate is first to tune them pure and then to make the necessary correction up or down. It is impossible in any other way to acquire true deli-

Practical Tuning in Equal Temperaments. 113

cacy of ear. No other method will produce the re- quired results. This is an immensely important truth and covers one of the most neglected ele- ments in the art. The rule of tuning all intervals pure before proceeding to temper, is essential ; and the student ought early to achieve this habit. The mere fact that Equal Temperament is an artificial mathematical compromise, should in itself be suf- ficient to persuade the tuner that this view is sound, especially when the reasons adduced at the beginning of this chapter are recalled. Until one knows practically the true beauties of Pure Intonation, the almost equally strong virtues and vices of Equal Temperament will in neither case be duly appreciated.

Any one who has had the privilege of listening to a first-rate string quartet or to a really good unaccompanied chorus, will understand why pure intonation, once recognized, is ever after treas- ured in memory.

Chapter V.

MECHANICAL TECHNIQUE OF TUNING.

The art of tuning tlie piano comprises two dis- tinct and separate elements. That part of our education as tuners which relates to the science of the art has already been discussed in full in the previous chapters. We must now consider what may be called the general mechanical technique of the art, including the special subject of the tools required and their manipulation.

The Raw Material. The raw material with which the tuner works may be described for prac- tical purposes as consisting of the string, the wrest-pin or tuning-pin, the wrest-plank or pin- block, and the tuning hammer. The piano action, the digitals, the wedges for damping strings, and the tuning-fork may alike be considered for the present, as accessory to, rather than as part of, the essential material. Let us consider these in their order.

String Conditions. It is well known, of course,

114

Mechanical Technique of Tuning. 115

that the pitch of a string — that is to say, its fre- quency of vibration — varies as the square root of its tension. Hence, if the tension be increased, the frequency is increased likewise; whilst if the tension be relaxed, the pitch is lowered. Now, the tension of a string is increased by tightening it on its pin ; that is by turning the pin so as to stretch the string more tightly. The relaxation of tension is achieved in precisely the opposite way ; that is, by releasing the string somewhat.

String and Pin. The piano string is wound so that the pin lies, as seen from the front, to its right, on an upright piano and to its left on a grand. But actually the positions are the same, and the differ- ence stated is due to the different position from which we view the strings on an upright and a grand respectively. It would be better to say that the string is always on the treble side of the pin in a horizontal and on the bass side in an upright piano. When the tuner wishes to raise the pitch of a string he turns the pin so as to wind up the string on it. To lower the pitch he turns the pin so as to unwind the string.

The mechanical problem therefore is to turn the pins in the wrest-planh in such a way as to adjust the pitch of each string to the requirements of the

116 Modern Piano Tuning.

Equal Temperament at the standard of pitch agreed on.

The Elements. In following chapters, I give a general description of piano construction and necessarily include remarks on the functions of all the parts herein mentioned. In the present chap- ter, however, I shall treat these only with relation to the tuner's work.

The String. The piano string varies in length inversely as its pitch and may be from 2 inches to 80 inches long, according to its position in the scale and the size of the piano. Steel music wire is used, varying in diameter from .03" to .06". The lower strings are wound with steel or copper overwinding or covering. The tension at which the strings are stretched varies within the limits of 100 and 275 pounds ; but it would be fair to name as an average range of tension per string in mod- ern pianos 150 to 160 pounds, although there is much inequality as to details.

Bearing. The opposite end of the string is passed around a hitch pin in the iron frame. In order to transmit its vibrations to the sound-board the string passes over a wooden bridge provided with double raked pins, so as to give it a side bearing. At the end near the tuning pin the string

Mechanical Technique of Tuning. 117

passes over a wooden or iron bearing bridge which determines the upper, as the sound^board bridge determines the lower, extremity of its vibrating length. This upper bridge gives an up and down bearing to the string. It is important that these points be kept in mind and the student carefully study the construction upon an actual piano; which I am presuming he has at hand.

The Pin Block. The torsional stress on the pin being what it is, considering the high tension at which the string is commonly stretched, it follows that efficient means are required for the main- tenance of the pin in a given position under this stress. The common method is to drive the pins into a wooden block called the wrest-plank or pin- block; which is built up by cross-banding several strips of hard wood. Such a block, when drilled for the reception of the tuning pins at right angles to the plane of the banding, gives an exceedingly stiff bedding, for the cross-grained strips present alternately end-grain and cross-grain, making a structure which interposes a very great frictional resistance to the rotative movement of the pin. It is the frictional resistance which is responsible for the pin holding its position, and hence for the string remaining at a given tension.

118 Modern Piano Tuning.

The Tuning-Pin. The tuning-pin is a stout steel rod almost uniform in diameter from end to end, and threaded with a light fine thread. One ex- tremity is bluntly pointed and the other is squared off to receive the tuning-hammer and pierced for the insertion of the end of the string. It is custo- mary to wrap the string around the pin in three or four coils.

The Bridges. The sound-board bridges carry the strings between pins set at an angle to each other, in such a way that each string is diverted from its line of direction and carried on from the bridge to the hitch-pin on a line parallel with the original line. The side-bearing thus given to the string is intended to ensure its tightness and steadiness on the bridge.

The upper bearing bridge sometimes is in the form of a separate stud or ''agraffe" for each string, and sometimes consists of a ledge cast in the plate over which the string passes, to be forced down into a bearing position by a heavy pressure bar screwed over it. The object is to give bearing to the string. The ''capo-d 'astro" bar is simply the pressure-bar arrangement cast in one piece. The student will be well advised to study all these constructions on the piano at first hand.

Mechanical Technique of Tuning. 119

The Tuning -Hammer. The tuning-hammer, with which the actual turning of the pin is accom- plished, is a steel rod carrying a head bored to receive the pin. The hammer is placed on the pin. which is turned by pressure of the hand on the hammer handle in the required direction. There are many interesting points about the manipula- tion of the hammer, however, which must now be considered.

The mechanical problem relates to the turning of a pin acting under a combined tensional and torsional strain; in other words a pin which is being simultaneously pulled down, and twisted around. If the wrest-plank is well made, the pin will resist successfully both of these strains and when turned by the hammer will retain its new position. But in order that this should be so it is necessary to acquire a certain technique of manip- ulation.

Manipulating the Hammer. The implement used for turning the pins and known as the tuning- hammer is, by its very shape, susceptible of wrong use. It presents the constant temptation to manipulate it as if it were a wrench. The resist- ance that the pin and string impose against turn- ing is sufficiently great to cause the novice nearly

120 Modern Piano Tuning.

always to twist and wrench at the pin in the effort to turn it. Now, it must be remembered that the distance through which the pin is turned is so very slight that where it has been tightly driven, or presents any other obstacle to free turning, the hammer nearly always gives too hard a twist or pull, so that the pin, after sticking, turns too far. In an upright piano the tuner stands in front of the pins, which are about on a level with his chest, and his natural inclination of course will be to pull downwards and outwards on the pins whilst at- tempting to turn them, in such a way as to drag the lower surface of the pin against the bushing in the plate and so gradually wear away the bush- ing and the wrest-plank hole and finally loosen the pin altogether. In order to overcome these possi- ble faults, the student will have to work out his own method of manipulation, bearing in mind al- ways that his object is to turn the pin and not merely bend it. If he merely bends it he may al- ter the string tension enough to change the pitch as desired ; but a smart blow on the piano key will soon knock the string back again where it was be- fore. That is why young tuners do not tune ** solidly." They do not turn the pins; they merely bend them. I shall offer the following sug-

Mechanical Technique of Tuning. 121

gestions regarding the general handling of the piano in tuning, out of my own experience, with the understanding that I do not put them forth as rules, but only as notions which have been found practical in one man 's work.

1. Length of Hammer. Other things being equal, the experience of the best tuners points to the use of a short handled hammer, instead of a long one. I prefer a handle not more than 12 inches long.

2. Position of Hammer. The hammer should be held nearly vertical when tuning upright pianos, and it is well to rest the arm, as this tends to give better leverage.

3. Turning the Pin. In attempting to turn the pin, do not jerk the hammer back and forth, nor on the other hand, use it like a wrench, but rather try to turn the pin by gently impelling it in the de- sired direction, feeling it all the time under the hand and avoiding the mistake of pulling down on the pin whilst turning it in the sharp direction.

4. Use of Left Hand, in upright piano. The best way of making sure that the pin is not pulled downwards whilst the string is being sharped is to hold the tuning hammer in the left hand. The pin is then raised in the process of sharping ; which

122 Modern Piano Tuning.

is as it should be. In any case, the hammer should be held vertical, or still better, inclining slightly over to the bass side. The above applies to the upright piano only.

5. And in Grand Pianos. In a grand piano, to tune with the right hand is best on account of the position of the tuner with relation to the strings ; which is opposite to the position with reference to the upright. But the highest treble strings, on ac- count of the peculiar construction of the grand piano, are most conveniently tuned with the ham- mer held in the left hand.

6. Tuning Pure Intervals First. All intervals that are to be tempered should be first tuned pure and then raised or lowered.^ Other things being equal, it is better to tune slightly above the re- quired pitch and let the string slack back; which can be assisted by a smart blow on the key. Coax- ing the string up to pitch usually involves its slack- ing off as soon as the piano is played.

7. Strings Hanging on Bridges. Strings often hang on the belly bridge and on the upper bearing. The waste ends at either extremity sometimes cause this trouble. The tuner must acquire the habit of so tuning that the string is pulled evenly

1 See page 96.

Mechanical Technique of Tuning. 123

through its entire length, from tuning-pin to hitch- pin, maintaining the tension of all its sections uni- form. To be sure that the pin is thoroughly turned gives the best assurance that the above re- quirement has been fulfilled.

8. '^ Pounding" Condemned. I do not believe in brutally pounding on the keys of a piano in an effort to "settle the strings." It is quite uncer- tain how much the strings can be * ' settled ' ' in any way like this ; and the process is objectionable in every other way.

9. Muting. The simplest way of muting the strings is by using a long strip of felt to stop off the outside strings of the triples and the alternate strings of the doubles, from one end of the piano to the other, before the tuning begins. Then the temperament Octave may be tuned on the middle string of each note and the Octaves up and down therefrom; after which the outside strings of the Unisons may be tuned all together. If however for any reasons such as those mentioned below, this causes the piano to stress unevenly, the Uni- sons can be adjusted section by section.

10. Position at the piano. All things considered it is better to stand up to tune all pianos, even grand pianos. The practice of sitting to tune up-

124 Modern Piano Tuning.

right pianos is certainly to be condemned, as it leads to slovenly handling of the hammer and general slackness.

11. Raising Pitch. In raising the pitch of a piano, go over it at least twice, the first time roughly, the second time smoothly. If the amount of rise required is very great the piano will need three tunings at once and another shortly after. Arrangements should be made with the owner of the piano in accordance with the extra amount of work to be done.

12. On Old Pianos. Raising the pitch on old pianos is always risky, as strings are likely to break. In emergencies, rust may be treated, at the upper bearing bridge and hitchpins, sparingly with oil. But oil is only to be used in emergencies, and with the utmost care to see that it does not reach the wrest-plank or soundboard bridges.

13. Lowering Pitch. In lowering the pitch, not less than three tunings will be required usually. This work is even more delicate than the above and needs even more care. The first tuning should be merely a rough letting down. Then the second may be a rough, and the third a smooth, tuning. But it is well to have an interval of a day between the second and the third tunings.

Mechanical Technique of Tuning. 125

14. Gang Mute. Even if a felt strip for a whole section is not used, it is advisable to have the entire temperament Octave wedged up whilst tun- ing it, so as not to have to disturb Unisons after they have been tuned. It is usually necessary to make various corrections in the temperament Octave as it is being tuned.

15. Uniform Pitch. Uniformity of pitch is a desideratum. Every tuner should have an inter- national pitch guaranteed fork, kept carefully in a felt-lined box and carefully guarded from rust. Tune to this pitch whenever possible. But do not make the mistake of trying to adjust all pianos to one pitch. It cannot be done.

16. Theatre Pianos. In tuning pianos for use in theatres with orchestras, it is advisable usually to tune a few audible beats above the pitch of the instrument which is used as a standard. This in- strument is usually the clarinet or cornet (in sym- phony orchestras, the oboe), which rises in pitch as it warms in the course of playing.

17. False Beats. The worst single enemy the tuner has is the string with false beats. In a good piano factory such strings are taken out and replaced before the piano leaves the premises. Sometimes faults in the scale, and especially the

126 Modern Piano Tuning.

fault of uneven tension, make for strings that beat when sounded alone. This beating arises through sections of the strings being unevenly stressed, whereby the corresponding partial tones are thrown out of tune. Such uneven stress may be the result of a twist put in the wire during the stringing, or of uneven thickness of the wire. When false beats are encountered, sometimes the tuner will find he can neutralize the beats by tun- ing the string slightly off from the other two of the triple. If no such expedient will work, then the strings must be left alone. Such false beats are especially to be found in the upper treble.

18. Bass Tuning. The real fundamental tones of the lowest strings are not actually heard. We hear instead upper partials thereof. Hence it is very often impossible to tune bass octaves by mere audition of beats between coincident partials. In this condition of affairs the tuner may tune by testing the Tenths, which is a good plan, or by isolating some partial and testing it with the cor- responding note above. To tune a clear bass is sometimes impossible.^

19. Test Intervals. All the tests recommended in the previous chapter should be used constantly

1 Cf. the discussions in Chapters II, III and IV.

Mechanical Technique of Tuning. 127

during the progress of the work, for the tuning will not be good otherwise. The most prolific source of imperfection lies in the accumulation of exceedingly slight errors; which soon mount up to intolerable mistunings. Constant testing, note by note, is therefore absolutely essential.

20. Sharp Treble. The temptation to tune the treble tones too high is one constantly to be avoided, for it is constantly present. Careful testing will alone rid the tuner of this error, which is insidious and habit-forming.

21. Lastly: Not twenty, but a hundred and twenty, rules or suggestions like these could easily be laid down ; but I prefer to leave the subject here. The student will learn at least one more : the value of Patience.

''Style." The student will also determine for himself, as time goes on, the characteristics of what may be called his special ''style." Tuning is an art and one which suffers more through being misunderstood than through any other single condition. The fine tuner is an artist in every sense of the word and the mental charac- teristics he must possess are such as not every- body can hope to have.

Understanding and Patience. Understanding

128 Modern Piano Tuning.

and Patience are the foundation of the tuner's art and for my part I am not sure to which of these qualities I should award the primacy. Certainly it is true that without Understanding the tuner is groping in the dark ; whilst without Patience he is already condemned in advance to failure in the attempt to do artistic work.

Experience. Experience too, is vastly impor- tant. The most talented student of the art finds that there is a mechanical technique to be mas- tered in tuning, just as in playing the piano. The necessary delicacy of wrist and arm, the neces- sary intuitive feeling that the pin has been turned as it should be; the necessary exquisite delicacy of ear : ^ all these faculties are the product of patient experiment and practice. The novice can- not expect to possess them; nor can he have any reason for being disappointed when he finds, as he will find, that to gain anything worth gaining, one must work — and work hard.

1 "Delicacy" here means rather "power of discrimination" than mere intuition. What is usually called a "musical ear" is noth- ing more than, at best, a feeling of tonality sometimes extending so far as unaided recognition of individual tones and tonalities on hearing them; and at worst, an inclination towards simple mel- ody, harmonically bare. Tlie tuner's audition is acoustical, not artificially musical. The ordinary "musical car" is of little value to him.

Mechanical Technique of Tuning. 129

< t

'Playing the Game." Still, it also remains that when one does take the trouble to play the game as it must be played, the reward is certain — and by no means contemptible; even when measured by mere money, the lowest of standards. On the whole, good tuners are as scarce to-day as they were fifty years ago. The tuning of every day is not good, usually; and the artistic tuner will find a hearty welcome, and adequate compen- sation, almost anywhere.

Finally: All that can be taught by a book I have here set forth. But mastery comes only through experience, combined with patient study and application. Various matters incidental to the art which do not come within the scope of *' tuning" proper are treated in the following chapters of this book.

Chapter VI.

THE MODERN PIANO.

Scope of this Chapter. The piano is the most familiar of musical instruments and one of the most accessible for purposes of examination. In the following pages, I shall assume that the reader has a piano at hand and will follow me through- out, with it as a model. Such a method will be found more satisfactory than if I asked the stu- dent to follow me with the aid merely of a few illustrations. I shall not undertake any critical examination of the details of piano construction, for this is the province of a special technical treatise,^ but shall confine myself to describing the features of the modern piano in a manner cal- culated to be of the greatest assistance to the tuner and repair man.

An Instrument of Percussion. The piano is a stringed instrument; and to this extent belongs

1 "Theory and Practice of Pianoforte Building," by the present writer.

130

The Modern Piano. 131

to the same general family as the violin. But its strings are excited by blows inflicted directly by a hammer and indirectly, through a mechanism called the '^ action," by the performer's hand. Hence the piano is also an instrument of percus- sion and belongs to the same general family with the dulcimer, the xylophone and the drums. This last fact is of great importance, for it is impossi- ble to understand the peculiarities of the piano unless we entirely forget its incidental likeness to other stringed instruments and concentrate our ideas upon the outstanding fact of percussion as the cause of the sounds evoked by it.

Upon the fact that the strings are violently struck, instead of being bowed or plucked, rests the entire character of the piano, making any com- parison of it with the violin or other stringed in- strument absurd. This is especially true with regard to the sound-board.

Three Elements. The piano proper comprises three elements ; the scale, the sound-board and the hammer-action. All other parts are entirely in- cidental and accessory.

Scale. The scale of the piano consists essen- tially of the set of strings which are struck by the hammers. There are eighty-eight digitals in the

132 Modern Piano Tuning.

key-board of the piano and thus eighty-eight sepa- rate tones. The strings are grouped three to an unison throughout some five octaves of the range, and thence in double grouping (2 to a note) down- wards to the lowest bass. The last ten or twelve at the lowest bass extreme are usually single strings.

Now it will be understood that the strings of the piano increase in length and weight as the scale descends. The highest note on the piano (C7) is evoked by a string some 2 inches long, and this length rather less than doubles at each oc- tave descending, until the lengthening process is brought to a stop at the size limits of the piano. This lengthening increases the weight of the de- scending strings until at about five octaves below C7, it becomes necessary to shorten the remain- ing strings between this point and the extreme bass on account of the size limits mentioned above. The requisite slowness of vibration therefore must be had by over-weighting the shortened strings; which is done by covering them with iron or cop- per wire. This covered section is usually strung cross-ways over the treble strings and is there- fore called the overstrung section.

Now the immediate point is that the main-

The Modern Piano. 133

tenance of this enormous mass of steel wire stretched tightly between fixed points and cap- able of withstanding the hardest blows of the hammers, means that (1) the tension at which each string is stretched must be high and (2) that in consequence, an elaborate structure must be provided for the purpose of supporting this mass under this high tension. The pull on each string averages not less than 160 pounds, taking mod- ern pianos by and large, although I consider this too high for the best tonal results. The total tensional strain therefore is not less than 35,000 pounds on the average; and this strain must be borne by the supporting structure.

Upright Piano; Plate and Bach. The support- ing structure in the upright piano consists of (1) a relatively thin plate or frame of metal backed by a decidedly heavy and massive framing of wood. The shape and dimensions of the plate vary with the size of the piano and the peculiarities of the individual scale plan. The back consists of three or more wooden posts crossed at top and bot- tom by heavy planks. The top plank is faced on the surface nearest the strings by a specially prepared wooden block or '^wrest-plank" into which are driven the tuning-pins which fasten the

134 Modern Piano Tuning.

upper extremities of the strings and wluch the tuner turns when adjusting the pitch of the piano ; ''tuning the piano" as we say.

The iron plate covers the front of this massive wooden back, and the sound-board is fastened to the back with the iron plate over it.^

Grand Piano: Plate and Back. The support- ing structure of the grand piano is somewhat dif- ferent. In the upright the sides and the casing generally are simply attached to the fundamental structure known as the back. But in the grand piano the whole case is glued around a rim of cross banded veneers which in turn encloses another rim, into which runs the system of braces and struts which comprises what corresponds to the upright back and on which sound-board and plate are laid.

First-hand study of pianos in grand and upright form will reveal all these matters to the student clearly.

Sound-Board and Bridges. The sound-board of the piano is the resonating apparatus which amplifies and modifies the string-sounds, so as to endue them with the characteristics of piano

1 The method of gluing the souiid-board to the back and all similar technical details may be found described in my "Theory and Practice of Pianoforte Building."

The Modern Piano. 135

tone. It is an open question how much influence the sound-board exercises in the development of tone. My own theory has been for long that the sound-board is a true vibrator and the direct producer of the piano tone; and that the string acts rather as the selector, imposing upon the board the particular wave-form which its own vi- bration evokes.^

From the tuner's point of view, the chief pres- ent interest of the sound-board lies in its physical character and its behavior under use. Even a cursory examination of the sound-board and of the bridges which cross it carrying the strings, indicates clearly that two essential conditions must exist if the board is to perform its resonat- ing duty well. The strings must be maintained on an adequate up-bearing and side-bearing, whilst the board must be in a state of tension. The board must be resilient, but also stiff, in a sense. It must be arched upwards to maintain itself against the immense down pressure of the strings, but also it must be built so that the neces- sary arching will have no injurious effect upon the wood-fibres, with consequent splitting or crack-

1 This matter is further treated in the following chapter. Cf. also, "Theory and Practice of Pianoforte Building," pp. 58 et seq,

136 Modern Piano Tuning.

ing. It will easily be seen then that the sound- board of the piano is a structure of exceeding deli- cacy, called upon to perform difficult and labori- ous duties.

The strings are carried over the sound-board on two wooden bridges, one for the overstrung bass and one for the remaining strings. It is customary to construct these bridges of cross- banded hard maple veneers, to avoid splitting. The strings in their passage across the bridges are given side-bearing by means of suitably driven pins. These bridges are glued on to the sound- board and secured from the back thereof by means of screws.

Ribs. The sound-board is ribbed with strips of the same lumber (spruce) which is used for the body of the board. The object of ribbing is

(1) to facilitate the impartation of a proper curve, or arch (called usually the *^ crown") to the board,

(2) to impart tension to the board and (3) to strengthen it against the string-pressure. Ribs are usually 12 to 14 in number and cross the board diagonally, from the top of the treble to the bot- tom of the bass, side.

The above description applies equally well to grand or upright pianos.

The Modern Piano. 137

Hammers. The strings of the piano are excited by blows inflicted on them by what are called * 'hammers." The hammer consists of a molded wooden head covered with a special kind of felt, and varying in size and thickness from treble to bass, the heaviest hammers being of course those in the bass. Postponing a complete technical dis- cussion of the piano hammer for a later chapter we may here remark that important considera- tions are the position of the hammer with refer- ence to the point at which each string is struck, and the nature of the mechanism whereby the performer translates his desires into mechanical action upon the hammer.

''Touch." All that which is known by the gen- eral name of "touch" in relation to the playing of the piano means, ultimately, control of the piano hammer. A great deal of confusion on the part of tuners, not to mention musicians and the lay public, would be altogether swept away if only it were realized that in piano playing the control of the string's vibration, which of course, means control of the wave-form and hence of the sound-board vibration, and hence lastly of the volume and quality of the tone, is entirely a mat- ter of the hammer. The weight of the hanuner.

138 Modern Piano Tuning.

the nature of its material as to density, etc., and the arc of travel through which it turns, together with its velocity, are the controlling factors in tone production. This turning of the hammer in obedience to the depression of the piano key is the province of what is called the ^'Action" of the piano.

Action. The '* action" of the piano is the mechanism interposed between hammer and per- former. It consists essentially of a set of digitals or finger-keys, one for each tone in the compass of the piano, and an equal number of lever-systems, consisting of levers turning in arcs of circles, one to each key, operating the hammer. When the key is depressed, the hammer is thrown forward, trips before it touches the string, is carried to the string by its own momentum, and instantly re- bounds. The string is allowed to vibrate so long as the key is depressed.^

Damper Action. The so-called "damper" is a piece of molded wood faced with soft felt, which presses against the string, but is lifted away therefrom as soon as the key is depressed. The damper allows the string to vibrate freely until

1 Cf. Chapter VITI for complete discussion of these points.

The Modern Piano. 139

the key is released, when it at once falls back on the string and silences it.

For the purposes of artistic piano playing, how- ever, it is necessary very often to take advantage of the sympathetic resonance of the sound-board by allowing the tone emitted by one or several string-groups to be strengthened, colored and otherwise enriched by the simultaneous sounding of other strings whose fundamentals are true par- tials to the originally sounded strings. When the entire line of dampers is lifted from the strings and held away from them by suitable mechanism, this property of the sound-board comes into play, and the warm color thus imparted to the tone con- stitutes one of the most valuable elements in piano playing.

In order to permit this advantage, the line of dampers is adapted to be pushed back from the strings (or in the grand piano, lifted up from them) by means of a rod actuated by a simple lever system which terminates in a ''pedal" op- erated by the right foot of the performer. This lever system is called the "trap-work" of the piano and is situated under the keyboard in the grand piano, with the pedals arranged in a frame-

140 Modern Piano Tuning.

work known as the *'lyre"; whilst in the upright the pedal and trap-work are placed at the bottom of the piano on what is called the '' bottom-board." The pedal is convenient to the performer's right foot and is called the "sustaining pedal"; some- times, wrongly, the "loud pedal."

Soft-Pedal. In the grand piano the keys and action are put together on a frame which can slide transversely. By depression of a second pedal the action is slid towards the treble, so that each hammer strikes only two strings of each triple, and one of each double, group. The effect is to soften the tone and modify its color. Similar trap-work is used between pedal and action, placed alongside the sustaining pedal action. The sec- ond pedal is placed conveniently to the per- former's left foot and is called the "soft pedal. ' '

In the upright piano the arrangement is the same except that instead of shifting the action, the hammers are pushed forward closer to the strings by a rod which rotates the rail against which the hammer-shanks rest.

Sostenuto or tone-sustaining pedal. On grand pianos, and occasionally on uprights, is to be found a third pedal situated between the other two

The Modern Piano. 141

and arranged to hold up dampers which are al- ready lifted by the action of the performer in play- ing on the keys ; and to hold them up as long as de- sired. A rod is rotated when the pedal is de- pressed, which catches against felt tongues on any dampers that may have been raised, and holds them up. So if the performer wishes to sustain a chord after his fingers have quitted the keys, he depresses this pedal after he has struck the keys and allows the strings to vibrate accordingly un- til the pedal is released.

Middle pedals in upright pianos are sometimes arranged for the same purpose, and sometimes lift the bass section of dampers. Sometimes they operate a ''muflBer," being a strip of felt that can be thrown between the hammers and the strings for the purpose of '' muffling" the sound. This is useful for practice purposes.

Case Work. The case of the upright piano con- sists of the following parts :

Sides : Glued on to the sides of the back.

Arms: Extending from sides to support key- bed.

Key-Bed : Upon which the key- frame is laid.

Toes : Extending from bottom of sides to sup- port Trusses.

142 Modern Piano Tuning.

Trusses : Resting on Toes and holding up key- bed. Sometimes called the '4egs."

Fall-Board: The folding lid over the keys. The double folding type is now usual and is called the "Boston" fall-board. The older type or single lid is called the "New York" fall-board.

Shelf: Laid over fall-board to support music.

Name-Board: Eesting over keys to support single type fall-board.

Key-slip: Strip in front of keys.

Key-Blocks: Heavy blocks at each extremity of keyboard.

Top-Frame: Folding or fixed frame, often elaborately decorated, which supports music and conceals piano action and hammers.

Bottom-Frame: Similar frame to the above, covering trapwork and parts of piano under key- bed.

Pilasters : Decorative pillars sometimes placed on either side of top-frame to support it.

Top: The folding lid which covers top frame and finishes off the casework of the piano.

Bottom-Rail: Rail running across the bottom of the casework, in which the pedals are housed.

Bottom-Board: Board on which trap work is mounted, behind the bottom-frame.

The Modern Piano.

143

Names of External Parts. Although the piano is such a familiar article there is a great deal of confusion as to the names to be applied to its

Figure 15,

1. Top.

2. Side.

3. Pilaster.

4. Top Frame, 6, Shelf.

6. Arm.

7, Key-Block,

8. Key-Slip.

9. Key-Bed.

10. Bottom Frame.

11. Bottom Rail.

12. Toe.

13. Truss.

14. Fall Board.

22 j 21-

[)riJiiiiiiiiiiiiiii(iiiiiiiiiJiJiiiiiiiiii!!iiiii(iiiiiiiani)i))iii!)ifiiii[iH^^^^^

^ijiiiiiiiiiiiiiiiiiiiiiiini/jiiiiiiiiiiiiiiiiiniiiiiiiiiimimnniiiimiii I i m «i,

jiiiiSH iniiininiiiHiiii

III I I I I I I I I I ( M I I I I I I M I M r I I I I I II M M I I I I I I I I I I I I I r4^

-tt>

19 I9a 19b

Figure 16.

1. Top.

2. Iron Plate, covering wrest

plank.

3. Treble tuning-pins,

4. Side.

5. Muffler-rail and muffler

felt.

6. Hammers.

7. Ilammer-rail.

8. Action.

9. Arm. '

10. Digitals or Keys.

11. Key-Bed.

12. Truss.

13. Toe.

14. Sound-board.

15. Iron Plate.

16. Sustaining Pedal. 144

The Modern Piano. 145

17.	Muffler Pedal,	20.	Muffler Trap-work,
18.	Soft Pedal.	21.	Action.
19.	Trap-work.	22.	Soft Pedal Lifter.
19a.	Bottom Kail.	23.	Action-bolt.
19b.	Bass bridge.	24.	Bass Tuning Pins.
19c.	Treble Bridge.

various parts. In figure 15 are shown the ex- ternal or visible parts of an upright piano with the proper name for each appended.

The various case parts of the grand piano are of course largely similar, but the different posi- tion of the scale necessitates modifications, which involve some changes in names and positions of parts, as shown on page 147.

Names of Internal Parts. In order to provide the reader with a list of correct names for the vari- ous internal parts of the piano, the illustration on page 144, figure 16, shows an internal front view of an upright piano.

The rear view shown on page 146 indi- cates the position of the various elements in the back frame and the rear of the sound- board.

The grand piano, figure 18, page 147, cannot so well be shown as its internal parts are hidden by the solid case. The differences, such as they are, which exist between the two types, however, are thoroughly explained in the pages which follow.

Figure 17.

1. Top Block of Back, behind wrest-plank.

2. Limiting Rim of Sound-board.

3. Posts.

4. Ribbing.

5. Surface of Sound-board.

6. Bottom-rail of Back.

7. Limiting rim of Sound-board.

146

		Figure 18,
1. 2. 3. 4. 5. 5a.	Top. Top-Stick or Prop. Case. Key-Bed. Leg. Pedal-Frame or Ly	6. Pedal Frame Brace 7. Pedal Rod. 8. Key-Slip. 9. Key-Block. 10. Fail-Board. re. 11. Music Desk.

147

148

Modern Piano Tuning.

Materials Used in Piano Construction.

Woods : Mahogany. Walnut. Oak.

Circassian Walnut. Bird's Eye Maple.

Maple.

White Wood.

White Pine.

Spruce.

Pear.

Holly.

Sycamore.

Cedar.

Mahogany.

Leiatheks : Doeskin. Elkskin. Buckskin.

Fext and Cloth:

Green and White Baize. Tone Felt. Damper Felt, hard. Damper Felt, soft. Flannel.

IVOBY.

Celluloid. Iron.

Steel.

Bbass. Gbaphite.

Used In

Veneers for Cases.

Veneers for Cases.

Veneers for Cases.

Veneers for Cases.

Veneers for Cases.

Veneers for Cases.

Wrest-Planks.

Backs.

Hammer moldings and shanks.

Hammer rails, dowels, etc.

Body of case work.

Key Frames and keys.

Soimd-boards.

Various small action parts.

Various action parts.

Key-rail cloth, punchings.

Hammers.

Bass dampers.

Treble-dampers.

Casework punchings, fall-board

8tri])s, name-board strips,

stringing. Tops of white keys.

Fronts of white keys.

Iron plate, action brackets, lx)lts and general hardware.

Action angle rails, plates, trap- work springs, etc. Action-springs, pedal feet, rods,

Lubrication of action, etc.

The Modern Piano. 149

Various other materials are used in small quan- tities, for individual manufacturers have their own special methods which require special mater- ials. But the above includes the principal ma- terials common to all pianos.

Finish. Modern pianos are elaborately finished with a highly polished surface. The base of this finish is several coats of varnish, which are rubbed down and then re-varnished with what is called a ''flowing" coat of very heavy varnish. This is again rubbed down, first with pumice stone, felt pad and water, then with rotten stone, felt pad and water and then with the hand. The final finish is given by rubbing with lemon oil, which is lastly wiped off with cheese cloth wrung out in alcohol.

Although this finish is very beautiful it does not retain its brilliancy long under domestic condi- tions. In the remarks on piano repairing I have made several suggestions concerning the repair of damaged varnish work.

This brief description of the modern piano has been intended to furnish the student only with an explanation of the relation of the various parts to each other and the correct functions of each. More thorough studies are made in following

150 Modern Piano Tuning. /

chapters of certain elements which the tuner re- quires to understand in completeness and the present chapter will be perhaps most useful in providing a convenient peg on which to hang them. Although it fulfills so humble a purpose, however, it will not be without its value if it im- presses on the reader's mind the great truth that the piano, as it stands, is by no means to be re- garded as the fruit of sudden inspiration but rather as the contemporary stage in a long proc- ess of evolution. The history of the instruments which preceded the piano in point of time, and which in system are its ancestors, shows plainly that the invention of the hammer action by Cristofori in 1711 was merely the culmination of a long series of efforts on the part of many great craftsmen, looking towards the production of a musical stringed instrument capable of doing for domestic use what the organ has always done for the church; namely, furnish complete command over all existing resources of harmony as well as of melody. The piano as it stands to-day is the crown of three centuries of endeavor; but it is by no means certain that it will not yet be modi- fied much further. No one can pretend that the piano is a perfect instrument. Its tempered in-

The Modern Piano. 151

tonation, its rather hard unmalleable tone, its lack of true sostenuto, all represent defects that must in time be improved out of existence. Meanwhile, we have to take the piano as we find it, realizing that after all it is a very fine and very wonder- ful instrument.^

Incidentally, it is a matter for congratulation that the modern development of the piano is almost wholly an American achievement ; and that European makers are confessedly inferior to the best of their American colleagues. Why this should be so is another matter; but it certainly is so.

1 The reader who desires to study the extremely fascinating liis- tory of the piano may find an extensive literature on the subject. Hipkins is the best authority by all means. See bibliographical note at the end of this volume.

Chapter VII.

SOUND-BOARD AND STRINGS.

Quite as characteristic of the piano's individu- ality as the hammer action itself, is the apparatus of resonance, or, as we more usually call it, the sound-board. The piano is a stringed instrument and thus claims kinship with viols and lutes and all their descendants ; but ever so much more it is a resonance instrument and a percussion instru- ment. In fact, the true character of the piano cannot be rightly apprehended until we have real- ized that the string-element is really overshad- owed to a considerable extent by the sound-board. The piano is just as much dependent upon reso- nance as upon the prior vibration of the strings. Without the sound-board the piano would have neither power nor color to its tones. Moreover, variations in the quality of the sound-board ma- terial in its construction and in the skill of its de- sign involve parallel variations in the tonal values of pianos, of such marked and distinct charac-

152

Sound-Board and Strings. 153

ter as, almost without any special physical in- vestigation, to convince us that we must accord to the combined tone-apparatus which we call the sound-board and strings, entirely individual pe- culiarities and functions.

In fact I propose in this chapter to consider the sound-board and strings together as one complete structure, which for want of a better term, we might name the ''tone-emission apparatus" of the piano. In this and in what follows, I do not wish to be considered dogmatic, and certainly have no intention of composing vague and involved disquisitions on the subject-matter. Practical throughout this book is proclaimed to be; but it is impossible to talk practically about the piano's sound-board and its strings, unless we have a solid basis of fact on which to found our theories. In- deed, in this particular case, as in many others, the one sure method of going astray is to rely on rule-of-thumb or traditionary notions; as the experience of numberless persons who have tried to improve the sound-board, most clearly, if pain- fully, indicates. Beginning therefore with a clear discussion of the phenomena seen in the action of the sound-board and the strings, I shall try to work out the bearing of these upon the facts of

154 Modern Piano Tuning.

piano construction as they affect the piano tuner in his work, the pianist in his playing and the piano in its durability and value.

Definition of tone-emission apparatus functions. The object of the tone-emission-apparatus may be described as follows: to produce the charac- teristic piano tone, through the vibration of the strings in response to the percussive action of the hammers thereon, and through the resonating functions of the sound-board, whereby the original string wave-forms are combined, amplified, and transformed in quality as required for the pur- pose indicated.

That is not a neat definition perhaps, nor is it uncommonly accurate in all its parts ; but for the present it is perhaps the truest description that can be assimilated. Later on we shall improve and refine the details with better understanding.

Piano Tone. The feature of the piano which distinguishes it generally from all other musical instruments, and specially from all other stringed instruments, is the peculiar character of its tone. This is, to an extent, of course, hard and un- malleable. It possesses neither the plasticity of the violin tone nor the bitter-sweet gayety and lightness of the guitar. It is solid, yet evanescent,

Sound-Board and Strings. 155

hard yet capable of infinite gradation in inten- sity. Lacking the serenity and majesty of the organ diapason, it is pre-eminent in obedience to touch. The pianist cannot indeed sustain his tones, nor swell or diminish them at will. Here both organ and violin surpass the piano. But the pianist can color his tone almost as widely as the violinist, and withal has a touch control over dynamics which the organ entirely lacks. Thus the tone of the piano, as brought forth by a good performer, has qualities highly attractive, which, combined with the convenience of the instrument, its capacity for complete musical expression in all possible harmonic relations, and its moderate price, have made it supreme in popularity. Let us then see just how this peculiar tone is pro- duced.

Acoustical Definition of Piano Tone. Speaking from the view-point which we have adopted in Chapter II, it may be said that piano tone is the effect of a wave-form induced by hammers strik- ing upon heavy high-tension stretched strings at pre-determined points on the surface thereof; these waves having definite forms which are modi- fied by the resonating power of the sound-board. The first important feature is that the piano tone

156 Modern Piano Tuning.

is produced by the strings being struck; thus dis- tinguishing the piano from all other stringed in- struments.

The string is struck. As we have already found out ^ a string stretched at high tension and struck by a piano hammer, is thrown into an extremely complex form of vibration. This vibrational form consists of the resultant of a number of simple forms, which in turn are the effect of the string's vibrating in various segments as well as in its whole length. In short, the fundamental tone of the string, together with partial tones correspond- ing to at least the following five divisions,^ sound together whenever the hammer makes its stroke. The exact number of concomitant partials depends, partly upon the amplitude of the vibration,